Difference between revisions of "Wavemaker Theory"

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A paddle with draft <math> D\, </math> is undergoing small amplitude horizontal oscillations with displacement
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{{Ocean Wave Interaction with Ships and Offshore Structures
 +
| chapter title = Wavemaker Theory
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| next chapter = [[Ship Kelvin Wake]]
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| previous chapter =  [[Wave Momentum Flux]]
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}}
  
<center><math> \xi (t) = \mathfrak{Re} \left \{ \Pi e^{i\omega t} \right \} </math></center>
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{{complete pages}}
  
Where <math> \Pi\, </math> is assumed known and real. This excitation creates plane progressive waves with amplitude <math> A \, </math> down the tank. The principal objective of wavemaker theory is to determine <math> A \, </math> as a function of <math> \omega, \Pi \, </math> and <math> H \, </math>.
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== Introduction ==
  
Other types of wavemaker modes may be treated similarly.
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[[Image:Wave_maker.png|600px|right|thumb|Wavemaker]]
  
In general, the wavemaker displacement at <math> X=0\, </math> may be written in the form
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We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]].
 +
A paddle is undergoing small amplitude horizontal oscillations with displacement
 +
<center><math> \zeta (z,t) = \mathrm{Re} \left \{\frac{1}{-\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} </math></center>
 +
where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that
 +
<math>f(z)\,</math> is real but this is not necessary.
 +
Because the oscillations are small the [[Linear and Second-Order Wave Theory| linear equations]] apply (which will be given formally below).
 +
This excitation creates plane progressive waves with amplitude <math> A \, </math> down the tank. The principal objective of wavemaker theory is to determine <math> A \, </math>
 +
as a function of <math> \omega, f(z) \, </math> and <math> h \, </math>. Time-dependent wavemaker theories can also be developed.
  
<center><math> \xi(t) = \mathfrak{Re} \left \{ \Pi (Z) e^{i\omega t} \right \} </math></center>
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== Expansion of the solution ==
  
Where <math> \Pi(Z) \, </math> is a known function of <math> Z \, </math>.
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{{frequency definition}}
  
Let the total velocity potential be:
 
  
<center><math> \Phi = \mathfrak{Re} \left \{ \phi e^{i\omega t} \right \} </math></center>
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{{velocity potential in frequency domain}}
  
where
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The equations therefore become
 +
{{standard linear wave scattering equations without body condition}}
 +
The boundary condition at the wavemaker is
 +
<center>
 +
<math>
 +
\left. \partial_x\phi \right|_{x=0} = \partial_t \xi = f(z).
 +
</math>
 +
</center>
 +
We must also apply the [[Sommerfeld Radiation Condition]]
 +
as <math>x\rightarrow\infty</math>. This essentially implies
 +
that the only wave at infinity is propagating away.
  
<center><math> \phi = \phi_\omega \ + \psi </math></center>
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{{separation of variables for a free surface}}
  
The first term is a velocity potential that represents a plane progressive regular wave down the tank with amplitude <math> A \, </math>, yet unknown. Thus:
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== Expansion in Eigenfunctions ==
  
<center><math> \phi_\omega = \frac{igA}{\omega} \frac{\cosh K (Z+H)}{\cosh KH} e^{-iKX + i\omega t} </math></center>
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The wavemaker velocity potential <math> \phi \,</math> can be expressed simply in terms of eigenfunctions
  
with:
+
<center><math> \phi = \sum_{n=0}^{\infty} a_n \phi_n (z) e^{-k_n x} </math></center>
  
<center><math> \omega^2 = gK \tanh KH. \,</math></center>
+
and we can solve for the coefficients by matching at <math>x=0 \,</math>
  
The second component potential <math>\psi\,</math> is by definition a decaying disturbance as <math> X \to \infty \, </math> and otherwise satisfies the following boundary value problem:
+
<center><math> \left. \phi_x \right|_{x=0} = \sum_{n=0}^{\infty} -k_n a_n  \phi_n (z) = f(z)
 
 
<center><math> \begin{cases}
 
  \nabla^2 \psi = \psi_XX + \psi_ZZ = 0, -H < Z < 0 \\
 
  \psi_Z - \frac{\omega^2}{g} \psi = 0, Z=0 \\
 
  \psi_Z = 0, Z=-H \\
 
  \psi \to 0, X \to \infty
 
\end{cases}
 
 
</math></center>
 
</math></center>
  
The condition on the wavemaker <math> (X=0) \, </math> is yet to be enforced.
+
It follows that
  
Note that unlike <math> \phi_\omega, \psi \, </math> is not representing a propagating wave down the tank so it is called a non-wavelike mode. Such modes do exist as will be shown below. On the wavemaker <math> (X=0) \, </math> the horizontal velocity due to <math> \phi_\omega\, </math> and that due to <math> \psi\,</math> must sum to the forcing velocity due to <math> \xi(t) \, </math>.
+
<center><math> a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z  </math></center>
  
Noting that <math> \phi_\omega \sim e^{-iKX} \cosh K(Z+H) \, </math> we will try <math> \phi \sim e^{-\lambda x} \cos \lambda (Z+H) \,</math>. Its conjugate which satisfies the condition of vanishing value as <math> X \to \infty </math> for <math> \lambda > 0 \,</math>.
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=== Far Field Wave ===
 
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One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude <math> A \, </math> in terms of <math> f(z) \, </math>.
<u> Laplace </u>: <math> \psi_XX + \psi_ZZ = 0, \, </math> verify for all <math> \lambda\,</math>.
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The far-field wave component representing progagating waves is given by:
 
 
<u> FS condition </u>: <math> \psi_Z - \frac{\omega^2}{g} \psi = 0 \qquad \qquad \Longrightarrow \quad - \lambda \sin \lambda H - \frac{\omega^2}{g} \cos \lambda H = 0 </math>
 
 
 
<center><math> \Longrightarrow \quad \lambda \tan \lambda H = - \nu \equiv \frac{\omega^2}{g} </math></center>
 
 
 
<u> Seafloor condition </u>: <math> \psi_Z = 0, Z=-H \, </math>
 
 
 
So for the non-wavelike modes <math> \psi, \lambda \,</math> must satisfy the "dispersion" relation
 
 
 
<center><math> \lambda \tan \lambda H = - \nu = - \frac{\omega^2}{g} < 0 </math></center>
 
 
 
For positive values of <math> \lambda \, </math> so that <math> e^{-\lambda X} \to 0, X \to + \infty \, </math>.
 
 
 
Values of <math>\lambda_i \, </math> satisfying the dispersion relation follow from the solution of the non-dimensional nolinear equation
 
 
 
<center><math> \tan \omega = - \frac{\nu}{\omega}, \omega = \lambda H \, </math></center>
 
 
 
Solutions <math> \omega_i, i = 1, 2, \cdots \, </math> exist as shown above with <math> \omega_i \sim i \pi \, </math> for large <math> i \, </math>. These values are known as the eigenvalues or eigen-wavenumbers of the non-wavelike modes. The eigen-wavenumber of the wavelike solution <math> K\, </math> is given by the dispersion relation:
 
 
 
<center><math> \frac{\omega^2 H}{g} = KH \tan KH. \, </math></center>
 
 
 
Verify that by setting <math> K = i \lambda \, </math>, the dispersion relation of the non-wavelike nodes follows. In summary the purely imaginary roots of teh surface wave dispersion relation and its single real positive root enter the solution of teh wavemaker problem.
 
 
 
Define teh following orthogonal eigenmodes in teh vertical direction <math> Z \, </math>:
 
 
 
<center><math> f_0 (Z) = \frac{\sqrt{2} \cosh K ( Z + H )}{{ (H + \frac{1}{v} \sinh^2 KH )}^{1/2}} </math></center>
 
 
 
<center><math> f_n (Z) = \frac{\sqrt{2} \cosh \lambda_n ( Z + H )}{(H + \frac{1}{v} \sinh^2 \lambda_n H )}, \qquad n = 1, 2, \cdots </math></center>
 
 
 
Selected to satisfy:
 
 
 
<center><math> \begin{cases}
 
  \int_{-H}^0 f_0^2 (Z) dZ = \int_{-H}^0 f_n^2 (Z) dZ = 1 \\
 
  \int_{-H}^0 f_m^2 (Z) f_n (Z) dZ = 0, \quad m \ne n
 
\end{cases} </math></center>
 
 
 
So the wavemaker velocity potentials <math> \phi_w \, </math> and <math> \psi\, </math> can be expressed simply in terms of their respective eigen modes:
 
 
 
<center><math> \phi_w = a_0 f_0 (Z) e{-iKX} </math></center>
 
 
 
<center><math> \psi = \sum_{n=1}^{\infty} a_n f_n (Z) e^{-\lambda_n X} </math></center>
 
 
 
and:
 
 
 
<center><math> \Phi = \mathfrak{Re} \left \{ ( \phi_w + \psi) e^{i\omega t} \right \} </math></center>
 
 
 
On <math> X=0 \, </math>:
 
 
 
<center><math> \Phi_X = \mathfrak{Re} \left \{ ( \phi_W + \psi_X)_X e^{i\omega t} \right \}</math></center>
 
 
 
<center><math> \frac{d\xi}{dt} = \mathfrak{Re} \left \{ \Pi (Z) i \omega e^{i\omega t} \right \} </math></center>
 
 
 
Or:
 
  
<center><math> \frac{\partial}{\partial X} (\phi_W + \psi)_{X=0} = \Pi (Z) i \omega </math></center>
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<center><math> \lim_{x\to\infty} \phi = a_0 \phi_0(z) e^{-k_0 x} =
 +
a_0 \frac{\cos k_0(z+h)}{\cos k_0 h } e^{-k_0 x} </math></center>
  
<center><math> \left. \frac{\partial\phi_W}{\partial X} \right |_{X=0} = a_0 ( -iK) f_0 (Z) </math></center>
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Note that  <math> k_0 \, </math> is imaginery. We therefore obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement <math> f(z) \, </math>.
 
 
<center><math> \left. \frac{\partial\psi}{\partial X} \right |_{X=0} = \sum_{n=1}^{\infty} a_n ( -\lambda_n) f_n (Z) </math></center>
 
 
 
It follows that:
 
 
 
<center><math> - i K a_0 f_0 (Z) + \sum_{n=1}^{\infty} a_n (- \lambda_n) f_n (Z) = i \omega \Pi (Z) </math></center>
 
 
 
One of the primary objecives of wavemaker theory is to determine <math> a_0 \, </math> (or the far-field wave amplitude <math> A \, </math> ) in terms of <math> \Pi (Z) \, </math>. Multiplying both sides by <math> f_0 (Z) \, </math>, integrating from <math> - H \to 0 \, </math> and using orthogonality we obtain:
 
 
 
<center><math> - i K a_0 = i \omega \int_{-H}^0 dZ f_0 (Z) \Pi (Z) </math></center>
 
 
 
<center><math> \Rightarrow \quad a_0 = - \frac{\omega}{K} \int_{-H}^0 dZ f_0 (Z) \Pi (Z) </math></center>
 
 
 
The far-field wave component representing progagating waves is given by:
 
  
<center><math> \phi_w = a_0 \frac{\sqrt{2} \cosh K (Z+H)}{{\left( H+\frac{1}{v} \sinh^2 KH \right)}^{1/2}} e^{-iKX} </math></center>
+
For what type of <math> f(z) \, </math> are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that
  
<center><math> \equiv \frac{igA}{\omega} \frac{\cosh K (Z +H)}{\cosh KH} e^{-iKX} </math></center>
+
<center><math> f(z) \ \sim \ \phi_0 (z) </math></center>
  
Plugging in <math> a_0\, </math> and solving for <math> A \, </math> we obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement <math> \Pi (Z) \, </math> and the other flow parameters.
+
Unfortunately this is not a "practical" displacement since <math> \phi_0 (z) \, </math> depends on <math> \omega\, </math>, so one would need to build a flexible paddle.
  
<u>Exercises</u>
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== Matlab Code ==
  
* Try <math> \Pi (Z) = A \begin{cases} 1, \qquad -D < Z < 0 \\ 0, -H < Z < -D \end{cases} </math> paddle-type wavemaker. determine the amplitude <math> A_W \, </math> and phase of the far-field wave-train.
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A program to calculate the coefficients for the wave maker problems can be found here
 +
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/wavemaker.m wavemaker.m]
  
* Repeat above exercise fro a hinge type wavemaker:
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=== Additional code ===
  
* For what type of <math> \Pi(Z) \,</math> are the non-wavelike modes <math> \psi \equiv 0 \, </math>? It is easy to verify by virtue of orthogonality that:
+
This program requires
 +
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
 +
to run
  
<center><math> \Pi(Z) \ \sim \ f_0 (Z) </math></center>
+
-----
  
Unfortunately this is not a "practical" displacement since <math> f_0 (Z,K) \, </math> depends on <math> K\,</math>, thus on <math> \omega\, </math>. So one would need to build a flexible paddle!
+
This article is based in part on the MIT open course notes and the original article can be found
 +
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/737E217E-0582-45D5-B1F9-B2ECF977C66E/0/lecture6.pdf here]
  
* What is the wave amplitude at infinity generated by a point source located at <math> Z = - D \,</math>?
+
[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]
  
More details on the above theory and extensions to the nonlinear case may be found in W&LAND MEI.
+
[[Category:Eigenfunction Matching Method]]
 +
[[Category:Pages with Matlab Code]]
 +
[[Category:Complete Pages]]

Latest revision as of 01:13, 5 May 2023

Wave and Wave Body Interactions
Current Chapter Wavemaker Theory
Next Chapter Ship Kelvin Wake
Previous Chapter Wave Momentum Flux



Introduction

Wavemaker

We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the Eigenfunction Matching Method. A paddle is undergoing small amplitude horizontal oscillations with displacement

[math]\displaystyle{ \zeta (z,t) = \mathrm{Re} \left \{\frac{1}{-\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} }[/math]

where [math]\displaystyle{ f(z) \, }[/math] is assumed known. Since the time [math]\displaystyle{ t=0 \, }[/math] is arbitrary we can assume that [math]\displaystyle{ f(z)\, }[/math] is real but this is not necessary. Because the oscillations are small the linear equations apply (which will be given formally below). This excitation creates plane progressive waves with amplitude [math]\displaystyle{ A \, }[/math] down the tank. The principal objective of wavemaker theory is to determine [math]\displaystyle{ A \, }[/math] as a function of [math]\displaystyle{ \omega, f(z) \, }[/math] and [math]\displaystyle{ h \, }[/math]. Time-dependent wavemaker theories can also be developed.

Expansion of the solution

We also assume that Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] and we assume that all variables are proportional to [math]\displaystyle{ \exp(-\mathrm{i}\omega t)\, }[/math]


The water motion is represented by a velocity potential which is denoted by [math]\displaystyle{ \phi\, }[/math] so that

[math]\displaystyle{ \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. }[/math]

The equations therefore become

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]


(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math]) The boundary condition at the wavemaker is

[math]\displaystyle{ \left. \partial_x\phi \right|_{x=0} = \partial_t \xi = f(z). }[/math]

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ x\rightarrow\infty }[/math]. This essentially implies that the only wave at infinity is propagating away.

Separation of variables for a free surface

We use separation of variables

We express the potential as

[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]

and then Laplace's equation becomes

[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]

The separation of variables equation for deriving free surface eigenfunctions is as follows:

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]

and

[math]\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }[/math]

We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha \, }[/math]

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by [math]\displaystyle{ k_{0}=\pm ik \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. The [math]\displaystyle{ k \, }[/math] of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

[math]\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }[/math]

to arrive at the dispersion relation

[math]\displaystyle{ \alpha = k\tanh kh. }[/math]

We note that for a specified frequency [math]\displaystyle{ \omega \, }[/math] the equation determines the wavenumber [math]\displaystyle{ k \, }[/math].

Finally we define the function [math]\displaystyle{ Z(z) \, }[/math] as

[math]\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

[math]\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }[/math]

Expansion in Eigenfunctions

The wavemaker velocity potential [math]\displaystyle{ \phi \, }[/math] can be expressed simply in terms of eigenfunctions

[math]\displaystyle{ \phi = \sum_{n=0}^{\infty} a_n \phi_n (z) e^{-k_n x} }[/math]

and we can solve for the coefficients by matching at [math]\displaystyle{ x=0 \, }[/math]

[math]\displaystyle{ \left. \phi_x \right|_{x=0} = \sum_{n=0}^{\infty} -k_n a_n \phi_n (z) = f(z) }[/math]

It follows that

[math]\displaystyle{ a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z }[/math]

Far Field Wave

One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude [math]\displaystyle{ A \, }[/math] in terms of [math]\displaystyle{ f(z) \, }[/math]. The far-field wave component representing progagating waves is given by:

[math]\displaystyle{ \lim_{x\to\infty} \phi = a_0 \phi_0(z) e^{-k_0 x} = a_0 \frac{\cos k_0(z+h)}{\cos k_0 h } e^{-k_0 x} }[/math]

Note that [math]\displaystyle{ k_0 \, }[/math] is imaginery. We therefore obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement [math]\displaystyle{ f(z) \, }[/math].

For what type of [math]\displaystyle{ f(z) \, }[/math] are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that

[math]\displaystyle{ f(z) \ \sim \ \phi_0 (z) }[/math]

Unfortunately this is not a "practical" displacement since [math]\displaystyle{ \phi_0 (z) \, }[/math] depends on [math]\displaystyle{ \omega\, }[/math], so one would need to build a flexible paddle.

Matlab Code

A program to calculate the coefficients for the wave maker problems can be found here wavemaker.m

Additional code

This program requires dispersion_free_surface.m to run


This article is based in part on the MIT open course notes and the original article can be found here

Ocean Wave Interaction with Ships and Offshore Energy Systems