European Journal of Sport Science, vol. 1, issue 1©2001 by Human Kinetics and the European College of Sport Science

A Model Analysis of the Effectsof Wobbling Mass on Whole-Body Vibration

Z. Yue; H. Kleinöder; Joachim Mester

The effects of wobbling mass on the whole-body vibration are studied in terms ofthe comparison between two models A and B regarding their detailed behaviorduring the "whole-body vibration," where Model A is a system of four degrees offreedom with rigid and wobbling masses in both lower body and upper body,while Model B is a system of three degrees of freedom with a rigid upper bodyand is otherwise identical to Model A. Various quantities are calculated for bothmodels. It is found that for the frequency range where the foot-to-upper-bodytransmission is important, the wobbling mass is able to reduce thetransmissibility, to reduce the amplitude of the oscillation of the center of mass ofthe body, and therefore to reduce the amplitude of the fluctuation of the externalforce from the source of vibration. The model analysis reveals the mechanism forthese reductions as follows: The oscillation of the wobbling mass in the upperbody is carried by the oscillation of the rigid mass of the upper body and thereforethe phase of the wobbling mass lags behind the phase of the rigid mass. For thisreason, the average power which the wobbling mass in the upper body gives tothe rigid mass of the upper body is negative.

Key Words: biomechanics, vibration, wobbling mass, model analysis, dynamicresponses

Key Points:

1. The effects of wobbling mass on the whole-body vibration are studied in termsof model analysis.

2. It is found that the wobbling mass is able to reduce the transmissibility, thevibration of the centre of mass of the body.

3. The wobbling mass is able to reduce the amplitude of the fluctuation of theexternal force from the source of vibration.

4. The mechanism of these effects is the phase shift between the wobbling and thecorresponding rigid masses.

IntroductionThe human body is exposed to vibration in many sports, for example, alpine skiing, mountingbiking (video: MPG format | RealPlayer format), and in-line skating (video: MPG format |RealPlayer format) (14, 24, 27), in some work performances (14, 35), as well as during travelingby road, rail, air, or sea.

The human response to vibrations involves not only biomechanical but also physiologicalprocesses. Biomechanics determines how the vibrations reach the body and propagate throughthe body, how each part of the body moves under given external vibration conditions, and so on,while physiology determines how the body reacts to these motions caused by the source ofvibration. Although a tremendous amount of experimental data concerning the seat-to-head orfoot-to-head transmissibility are available in the literature (e.g., see 14), the propagation of thevibration through the body is still poorly understood. As one example, the complication of thetopic can be seen by the fact that the frequency range of resonance, where the vibration isamplified rather than dampened, depends upon not only the part of the body, but also the positionof the body and the direction of the vibration (8, 9; also Table 1).

It is well known that vibrations can cause discomfort, for example, in various forms of travelsicknesses (by bus, boat, or aeroplane). Irvin’s sensory conflict theory (6) states that thediscomfort might be caused by a conflict between different sensory inputs that cannot bemanaged by the brain.

Vibrations also occur with some training machines (see the hydro pulser: MPG format |RealPlayer format)] and vibration training: video: MPG format | RealPlayer format). Theintention is to improve strength and flexibility without using heavy weights as in usual weighttraining (17, 21, 25, 31, 33, 36).

It is important to distinguish the vibration conditions for advantageous training from those forhealth risk. Although some statistical analysis along this line is available in the literature (16, 32;see Figure 1), the mechanism is still far from well understood yet.

Figure 1 — Health risk guidelines for different situations in everyday life and in sport (modified after ref. 32).

Obviously, to understand the biomechanical process is the basis of the understanding of thehuman response to the whole-body vibration. However, the biomechanical process itself is verycomplicated because the body consists of not only many segments but also of both rigid massesand wobbling masses.

Theoretical approaches have been various spring-damper-mass models, which were designed tosimulate either the whole-body vibration (cf. the conference proceedings: 1, 3, 4, and thereviews: 7, 12, 13, 20, 29, 30) or the running and hopping (2, 5, 10, 11, 18, 19, 22, 23, 26, 28,34). Although these models help to understand the propagation process of the vibration throughthe body to a certain extent, one caution should be made. Most of the models are designed to fitcertain data. These curve-fitting models may not be applicable to other research purposes.However, the other type of models, so-called concept models, which are designed not just to fitobserved data, but based on some physical or physiological reasoning, may have betterpredicting power to some other research topics under similar conditions.

The purpose of the present paper is to study the effects of wobbling masses on the whole-bodyvibration. We will use a slightly varied version of the model of Liu and Nigg (22; also Figure 2;Table 2 below) as a starting point because this model includes wobbling masses and wasdeveloped to study the effects of wobbling masses on running. Equally important is that thismodel was not designed just to fit existing data. Instead, the spring constants and dampingcoefficients in this model were determined, based on the consideration of some muscle-tendonproperties (22). Thus, this model belongs to concept models. We use this model to study theeffects of wobbling masses on whole-body vibrations for the case when the subject stands on thevibrating platform with one foot because the model was originally to simulate the impact duringrunning as one foot touches the ground. This model will be referred to as Model A. In order tosee the effects of the wobbling properties of the "internal organs" on the whole-body vibration,we will compare Model A with a Model B, which is the same mass as Model A.

Mathematical formulations of the "whole-body vibration" for Model A and Model B will begiven in Section 2 and Section 3, respectively. Results of calculations of various quantities forModel A and Model B will be presented and compared in Section 4. Implications of these resultswill be discussed in Section 5. Conclusions will be drawn in Section 6.

Formulation of the "Whole-Body Vibration" of Model AModel A (Figure 2) is a slightly varied version of the model used by Liu and Nigg (22). The partunder M1 in the original model for the simulation of the ground reaction during running is nowreplaced by the platform of the vibrator. We denote the vertical coordinate of the equilibriumposition of the center of mass of Mj as the system is at rest by Zj0, and the vertical coordinate ofthe center of mass of Mj as the system is vibrating by Zj (t) (j = 1,2,3,4). Thus, the deviations ofthe centers of mass of Mj (j = 1,2,3,4) from their equilibrium positions

j = Zj – Zj0 (j = 1,2,3,4) (equ. 2.1)

satisfy the following equations:

M1 1 = F – k2( 1 – 2) – c2( 1 – 2) – k1( 1 – 3) – c1( 1 – 3) (equ. 2.2)

M2 2 = –k2( 2 – 1) – c2( 2 – 1) – k3( 2 – 3) (equ. 2.3)

M3 3 = –k1( 3 – 1) – c1( 3 – 1) – k3( 3 – 2) – (k4 + k5)( 3 – 4) – c4( 3 – 4) (equ. 2.4)

M4 4 = –(k4 + k5) ( 4 – 3) – c4( 4 – 3) (equ. 2.5)

where • stands for the time derivative d/dt, and

F = Fp – Fp0 = Fp – (M1 + M2 + M3 + M4)g (equ. 2.6)

is the total external force acting on the system, in which Fp is the pressure the platform gives tothe system, Fp0 = (M1 + M2 + M3 + M4)g is the pressure of the platform in equilibrium state, g isthe acceleration of gravity, the masses Mj, spring constants kj, and damping coefficients cj aregiven in Table 2. The advantage of using j rather than Zj as the basic quantities for furtherdiscussions is that Equations (2.2)–(2.5) do not contain the parameters, for example, the naturallengths of the springs, explicitly. Actually, Equations (2.2)–(2.5) are derived from the subtractionof the equations for Zj0 from those for Zj (t), and those parameters are cancelled out during the

subtraction. Physically, j, j, and j stand for the displacement from the equilibrium position,the velocity, and the acceleration of Mj, respectively.

Figure 2 — Model A: The spring-damper-mass model used to simulate the whole-body vibrations in the presentstudy. This model as well as the parameters in Table 2 are essentially the same as those used in Liu and Nigg (22)except that the part under M1 to simulate the ground reaction in their paper is now replaced by a vibrator to whichM1 is assumed to be fixed. M1 and M2 simulate the rigid and the wobbling masses of the lower body, while M3 andM4 simulate the rigid and the wobbling masses of the upper body.

PRIVATETable 1 — Resonance Frequencies for DifferentParts of the Human Body in Different Body Positions andVarious Directions of Vibration (modified after ref. 16)Body position Body part Direction of

vibrationFrequency range

of resonance

FootKneeAbdomenRib cageCranial bone

xxxxx

16–31 Hz4–8 Hz4–8 Hz6–12 Hz50–70 Hz

FootAbdomenHead

yyy

Lying

FootAbdomenHead

zzz

0.8–3 Hz0.8–4 Hz0.6–4 Hz1–3 Hz1.5–6 Hz1–4 Hz

Standing KneeShoulderHeadWhole body

xxxz

1–3 Hz1–2 Hz1–2 Hz4–7 Hz

Sitting TrunkRib cageSpineShoulderStomachEye

zzzzzz

3–6 Hz4–6 Hz3–5 Hz2–6 Hz4–7 Hz20–25 Hz

PRIVATETable 2 — The Parameters of the SystemM1

(kg)M2

(kg)M3

(kg)M4

(kg)k1

(kN/m)k2

(kN/m)k3

(kN/m)k4

(kN/m)k5

(kN/m)C1

(kg/s)C2

(kg/s)C4

(kg/s)

6.15 6 12.58 50.34 6 6 10 10 18 300 650 1900

If we denote the positions of the center of mass of the system in equilibrium and vibrating statesby Zc0 and Zc respectively, the deviation c = Zc – Zc0 is then given by

c = Mj j (equ. 2.7)

where M = Mj is the total mass of the system. The summation of Equations (2.2)–(2.5) gives

F = Mj j = M c = M c (equ. 2.8)

which is the center-of-mass theorem of the system. The linear Equations (2.2)–(2.5) allow thesinusoidal solutions with a single frequency as follows:

j = ReAjeiω t (j = 1,2,3,4) (equ. 2.9)

F = Re eiω t(equ. 2.10)

where and Aj (j = 1,2,3,4) are the complex amplitudes of the corresponding quantities. FromEquation (2.7), we have

c = ReAc eiω t (equ. 2.11)

where

Ac = Mj Aj (equ. 2.12)

Only one of complex amplitudes could be assumed to be real. In the following, we will assumethat A1 is real, while Aj (j = 2,3,4), Ac and are all complex, namely

A1 = |A1| =A1R > 0, Aj = AjR + iAjI = | Aj | exp(I φj) (j = 2,3,4) (equ. 2.13)

Ac = AcR + iAcI = | Ac | exp(i φc), = R + i I = | | exp(i φF) (equ. 2.14)

where the complex moduli |Aj|, |Ac|, and | | represent the physical amplitudes of thecorresponding quantities, while the phase angles φj, φc, and φF are the phase shifts between thecorresponding quantities and the platform.

In order for the sinusoidal solutions to exist, it has also been assumed that M1 is always fixed onthe platform or, equivalently, that detachment never occurs. This can be assured if one of thefollowing two conditions is satisfied: (i) The frequency and the amplitude of the vibrator are notso large that the maximal acceleration of the center of mass of the system exceeds theacceleration of gravity, or (ii) the fixation of M1 on the platform is in both upward and downwarddirections so that detachment cannot occur.

Because of the linearity of Equations (2.2)–(2.5), it is convenient to look for the solutions of A2,A3, A4 for given A1. For this purpose, we introduce the following dimensionless complexquantities:

α1 = A2/A1, α2 = A3/A1, α3 = A4/A1, αF = /(ω2MA1), αc = Ac/A1 (equ. 2.15)

It is easy to show that aj (j = 1,2,3) are determined by the following linear algebraic equations:

Ωij αj = ηi (i = 1,2,3) (equ. 2.16)

where

Ω11 = (–M2 ω2 + k2 + k3)/k1 + i ω c2/k1 (equ. 2.17)

Ω22 = (–M3 ω2 + k1 + k3 + k4 + k5)/k1 + i ω (c1 + c4)/k1 (equ. 2.18)

Ω33 = (–M4 ω2 + k4 + k5)/k1 + i ω c4/k1 (equ. 2.19)

Ω12 = Ω21 = –k3/k1 (equ. 2.20)

Ω13 = Ω31 = 0 (equ. 2.21)

Ω23 = Ω32 = –(k4 + k5)/k1 – i ω c4/k1 (equ. 2.22)

η1 = k2/k1 + i ω c2/k1, η2 = 1 + i ω c1/k1, η3 = 0 (equ. 2.23)

From Equations (2.8) and (2.11), αF and αc are determined by αj (j = 1,2,3) as follows:

αF = –µ1 – µj + 1 αj (equ. 2.24)

and

αc = µ1 + µj + 1 αj = –αF (equ. 2.25)

where

µj = Mj / M (j = 1,2,3,4) (equ. 2.26)

Various interesting output average quantities, which can be expressed in terms of αj (j = 1,2,3),αF and the frequency, are summarized as follows:

(i) foot-to-upper-body transmissibility:

<( 3)2>1/2 / <( 1)

2> 1/2 = |A3| / A1 = |α2| (equ. 2.27)

(ii) apparent mass:

/(–ω2A1) = –αF M (equ. 2.28)

(iii) mechanical impedance:

/(iω A1) = –iω αF M (equ. 2.29)

(iv) dynamic stiffness:

/A1 = ω2αF M (equ. 2.30)

Apparent mass, mechanical impedance, and dynamic stiffness are all complex quantities.

(v) the amplitude of the total external force:

| | = ω2 |αF| M A1 (equ. 2.31)

Figure 3 — Model B: A system of three degrees of freedom with a rigid upper body of mass M3 + M4. Otherwise itis the same as Model A. All the parameters are still given in Table 2.

Formulation of Model BIf we simply remove the wobbling mass M4 and its connections (springs k4, k5 and the damperc4) in Figure 2 and merge the mass M4 into the rigid mass M3, we will get Model B (Figure 3).

Thus, Model B differs from Model A only in that the "upper body" in Model B is assumed to berigid and has the mass M3 + M4. We denote the deviations of the center of mass of M1, M2 andthe upper rigid body and the total external force for Model B by 1B, 2B, 3B, and FB, respectively.This system of three degrees of freedom is governed by the following equations:

M1 1B = FB – k2( 1B – 2B) – c2( 1B – 2B) – k1( 1B – 3B) – c1( 1B – 3B) (equ. 3.1)

M2 2B = – k2( 2B – 1B) – c2( 2B – 1B) – k3( 2B – 3B) (equ. 3.2)

(M3 + M4) 3B = – k1( 3B – 1B) – c1( 3B – 1B) – k3( 3B – 2B) (equ. 3.3)

where

FB = FpB – F = FpB – (M1 + M2 + M3 + M4)g (equ. 3.4)

is the total external force acting on the system. Similar to the discussions for Model A, we nowhave the corresponding equations for Model B as follows:

FB = M1 1B + M2 2B + (M3 + M4) 3B (equ. 3.5)

cB = [ M1 1B + M2 2B + (M3 + M4) 3B] (equ. 3.6)

jB = ReAjB eiω t (j = 1,2,3) (equ. 3.7)

FB = Re B eiω t (equ. 3.8)

cB = ReAcB eiω t (equ. 3.9)

AcB = [ M1 A1B + M2 A2B + (M3 + M4) A3B] (equ. 3.10)

where

A1B = |A1B| =A1B,R > 0, AjB = AjB,R + iAjB,I = | AjB | exp(i jjB) (j = 2,3) (equ. 3.11)

AcB = AcB,R + iAcB,I = | AcB | exp(i φcB), B = BR + i BI = | B| exp(i j FB) (equ. 3.12)

We introduce the dimensionless complex quantities:

α1B = A2B/A1B, α2B = A3B/A1B, αFB = B /(ω2MA1B), αcB = AcB/A1B (equ. 3.13)

The linear algebraic equations for the determination of a jB (j = 1,2) are as follows:

ΩijB αjB = ηiB (i = 1,2) (equ. 3.14)

where

Ω11B = (–M2 ω2 + k2 + k3)/k1 + i ω c2/k1 (equ. 3.15)

Ω22B = (–(M3 + M4) ω2 + k1 + k3)/k1 + i ω c1/k1 (equ. 3.16)

Ω12B = Ω21B = –k3/k1 (equ. 3.17)

η1B = k2/k1 + i ω c2/k1, η2B = 1 + i ω c1/k1 (equ. 3.18)

From Equations (3.5) and (3.10), αFB and αcB are determined by αjB (j = 1,2) as follows:

αFB = –[µ1 + µ2 α1B + (µ3 + µ4) α2B ] (equ. 3.19)

and

αcB = µ1 + µ2 α1B + (µ3 + µ4) α2B = –αFB (equ. 3.20)

For the output average quantities, we have:

(i) foot-to-upper-body transmissibility:

<( 3B)2>1/2 / <( 1B )2>1/2 = |A3B| / A1B = |α2B| (equ. 3.21)

(ii) apparent mass:

B/(–ω2 A1B) = –αFB M (equ. 3.22)

(iii) mechanical impedance:

B /(iω A1B) = –iω αFB M (equ. 3.23)

(iv) dynamic stiffness:

B /A1B = ω2αFB M (equ. 3.24)

(v) the amplitude of the total external force:

| B| = ω2 |αFB| M A1B (equ. 3.25)

ResultsFor the parameters shown in Table 2 and any given frequency f (ω = 2π f), all the dimensionlesscoefficients Ωij, ηi and ΩijB, ηiB are determined by Equations (2.17)–(2.23) and (3.15)–(3.18), respectively. The dimensionless complex amplitudes αj (j = 1,2,3) for Model A and αjB (j= 1,2) for Model B are determined by Equations (2.16) and (3.14), respectively. Thus, all theinteresting quantities can be calculated and compared for the two models. Some results areshown in Figure 4–11. Figure 4(a)–(d) show the amplitude ratios and phase shifts as functions ofthe frequency for Model A and Model B. Figure 5 gives the comparison of the foot-to-upper-body transmissibility between the two models. Figure 6 is the comparison of the amplitudes ofthe center of mass for the two models. Figure 7 shows the comparison of the amplitudes of thetotal external forces for the two models. Figure 8 (a)–(d) show the comparison of the dynamicresponses—the apparent mass, the mechanical impedance, and the dynamic stiffness—betweenthe two systems. Figures 9 and 10 show the detailed time changes of the displacements forModel A and Model B, respectively. Figure 11 gives the comparison of the force-time curves ofthe two models for two different frequencies. The implications of all these figures will bediscussed in the next section.

Figure 4 — (a) The amplitude ratios for Model A. (b) The phase shifts for Model A. (c) The amplitude ratios forModel B. (d) The phase shifts for Model B.

Figure 5 — Comparison of the foot-to-upper-body transmissibility between Model A and Model B.

Figure 6 — Comparison of the amplitude of the center of mass between Model A and Model B.

Figure 7 — Comparison of the amplitudes of the total external forces of Model A and Model B.

Figure 8 — Comparison of the dynamic responses between Model A and Model B: (a) apparent mass: modulus, (b)apparent mass: phase angle, (c) mechanical impedance: modulus, and (d) dynamic stiffness: modulus.

(cont.)

Figure 9 — Displacements vs. time at different frequencies for Model A: (a) f = 2.5 Hz, (b) f = 5 Hz, and (c) f = 10Hz.

Figure 10 — Displacements vs. time at different frequencies for Model B: (a) f = 2.5 Hz, (b) f = 5 Hz, and (c) f = 10Hz.

Figure 11 — Comparison of the force-time curves between Model A and Model B for (a) f = 2.5 Hz and (b) f = 5Hz.

DiscussionsFrom Figure 5 it is seen that the foot-to-upper-body transmissibility is reduced due to thewobbling mass for the frequency range in which the transmission is important. This reductioncan be understood in terms of the power analysis as follows: It is easy to show that the averagepower that the wobbling mass M4 gives to the rigid mass M3 (cf. Figure 2) is

<P43> = ω (k4 + k5) |A3| |A4| sin(φ4 – φ3) (equ. 5.1)

Since φ4 is smaller than φ3 [see Figure 4(b)], <P43> is negative. This explains why the amplitudeof the oscillation of M3 is reduced by the wobbling mass M4. The fact that the phase of thewobbling mass M4 lags behind the phase of the rigid mass can also be seen in Figure 9 (a)–(c)where 4 (t) reaches the peak always later than 3 (t) does. The physical reason is that thevibration of the wobbling mass M4 is carried by the vibration of the rigid mass M3. This is alsotrue for the real human body, in which all the internal organs are carried by the skeleton of theupper body, and therefore the phase of the wobbling mass lags behind the phase of the rigid massduring the whole-body vibration. Thus, we expect that the effects of the wobbling mass on thewhole-body vibration we get from the present model analysis are also true for the real humanbody, although the frequency range for large transmission is higher in the real human body [say,around 5 Hz, cf. Griffin (14)] than in the present model analysis. From Figure 6, it is seen thatthe amplitude of the oscillation of the center of mass of the system is reduced by the wobblingmass for the frequency range where the transmission is important. This explains why theamplitude of the oscillation of the external force is reduced by the wobbling mass in the samefrequency range (see Figure 7). Actually, from the center-of-mass theorem [equ. (2.8)], theamplitude of external force must be reduced if the amplitude of the center of mass of the systemis reduced, provided that the frequency and the total mass of the system remain the same.

ConclusionsThe present model analysis reveals the following effects of wobbling mass on the whole-bodyvibration: For the frequency range where the foot-to-upper-body transmission is important, thewobbling mass is able to reduce the transmissibility, to reduce the amplitude of the oscillation ofthe center of mass of the body, and therefore to reduce the amplitude of the fluctuation of theexternal force from the source of vibration. Since the oscillation of the wobbling mass in theupper body is carried by the oscillation of the rigid mass of the upper body, the phase of the

wobbling mass lags behind the phase of the rigid mass. For this reason, the average power thatthe wobbling mass in the upper body gives to the rigid mass of the upper body is negative. Thisis the basic mechanism for the above-stated effects of the wobbling mass.

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