Energy balance of the model as described in “Theory and Model”

1. Introduction

The energy balance as presented below is based on energy (or power) sources and –sinks and avoids such notions as power through the foot stretcher or through the oarlock.

My model does not contain internal accelerations and forces in the system and so I am obliged to differentiate boat and rower velocity. Differentiation in numerical simulation is in general not considered as good practice. It results in noise and spikes. Because in the end these signals are integrated again the noise disappears again.

The Simulink model used is the same as in “Theory and Model” only two differentiating blocks have been added. See Fig 1.1

2. Power sources and –sinks

The rower is the source of power. This source can be split into three sub sources: legs, back and arms. I order to keep it simple I consider legs and arms only. The rower is someone with a stiff back. Not ideal but for the time being acceptable, it is just the idea and the approach I want to make clear.

For me it is helpful to look at legs and arm as hydraulic cylinders that deliver energy by extending (legs) and contracting (arms) during the drive.

Now look at the rower model Fig 2.1 (I hope the reader can distinguish the anatomy of the rower):

The forces are the forces on the rower. The velocities are the velocities with respect to the boat (any other reference system will do because only velocity differences will appear in my reasoning). The acceleration a_rower of the rower’s mass m_rower is measured in a world fixed reference system.

Power delivered by the legs: P_legs = F_stretcher . v_seat

Power delivered by the arms: P_arms = F_{handle
.}(v_handle – v_shoulder)

Total power: P_tot = P_legs+ P_arms

This is all what is to say about power delivered by the rower. There is no need to introduce -stretcher or gate power.

(For further discussion on the power balance, see below)

Relation between F_stretcher and F_arms: F_arms = F_stretcher – m_rower.a_rower

power delivered = power dissipated + increase of kinetic energy (no potential energy is involved in rowing) for the situation that a blade is in the water.

power dissipated = power dissipated by fluid friction along the boat + power dissipated at the blade (= work done on the water by the blade force)

power dissipated = C.v_boat³ + F_blade . v_blade(this last term to be interpreted as a vector dot product)

C = a constant, v_boat = boat velocity with respect to the water and v_blade is the blade velocity with respect to the water.

The increase of kinetic can be written, in general as:

increase in kinetic energy = m_boat . v_boat . a_boat + m_rower . v_rower . a_rower

All velocities and accelerations with respect to an earth bound system = with respect to the water. Summarized, the power balance during the drive:

P_tot = C.v_boat³ + F_blade . v_blade+ m_boat . v_boat . a_boat + m_rower . v_rower . a_rower

3. Numerical results

In order to understand the following results is necessary to familiarise with the model. See “Theory and Model” and Results. The drag- and lift coefficients used follow the description of Caplan and Gardner. See the web page Lift and Drag.

******* initial data *********

m1 m2 Fbl L fi1 fi2 sl TR

kg kg N m rad rad m s

30.0 70.0 320 1.80 -1.10 0.60 0.60 1.10

area C1 C2

m2 N.s2/m2 -

0.130 3.00 1.20

------- results -------

Ebls Exs Et eff T

J J J - s

127.16 618.75 745.91 0.829 1.82

SR T2000 Prow Poar vel

m-1 s W W m/s

32.95 419.6 409.84 348.39 4.767

In the former simulation the inboard handle length was not specified. The present results are obtained with an inboard length Li = 0.9 m.

The amount of energy Ebls (energy dissipated at the blade) and Exs (energy outflow due to hull friction) has been summed to Et = 745.91 J. This follows immediately from the Simulink model.

In the present calculation the total energy derived from legs and arms = 744.88 J.

The energy calculated as outflow, including the storage as kinetic energy = 746.52 J.

These three numbers should be the same but the resemblance is satisfactory.

Fig 3.1 shows the normalised blade force. The maximum is set at 320 N.

Fig 3.2 shows the seat speed during the recovery.

In Fig 3.4 the accelerations are presented. Te green line is the acceleration of the common centre of mass and is directly produced by the Simulink model. The red and the green line are the products of differentiation. Some smoothing has been applied but the remains of some spikes at abt 0.75 s could not be removed.

In the following charts the title in the top may look rather cryptic. Consider them indications for the author. Under the charts an understandable title has been written.

Fig 3.5 displays the power generated by arms and legs and the power delivered at the oar that is not the sum of legs and arms power. Note that the legs’ power is considerable more than the arms’ power.

The difference is shown in Fig 3.7.

In Fig 3.6 the result of the integration of the power curves of arm and legs is shown.

Unfortunately, the colour codes are not the same as in Fig 3.6.

The above graphs all referred to the power (energy) sources arms and legs. The following graph shows the sinks of power, blade losses and hull friction, and the changes in kinetic energy in rower and hull.

The end points of the blue and the red sum up to the total energy consumption during one cycle. The zero level of kinetic energy is the situation at he start (and thus at the end) of the cycle. Therefore negative kinetic energies are possible.

Fig 3.9 shows the integrated power curves for sources and sinks. They cover each other almost perfectly as should be the case.

Final remarks

The presented results give a reasonable insight into the energy bookkeeping

during one cycle in the stationary situation.

The model of the rower as described in section 1 is not complete and perfect. A first refinement to be considered is a better description of the rower’s back. The purpose of the exercise was however to be as realistic as possible with simple means.

In contradiction to what was found in Energy balance in a simple model no situation has occurred where the “ engines” (power sources) absorb power instead of delivering it. Also Atkinson pays much attention to this “reversed” situation. This remains to be solved.

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