Update Content - 2021-04-14

content/phdthesis/rankers98_machin.md

@@ -8,7 +8,7 @@ Tags
 : [Finite Element Model]({{< relref "finite_element_model" >}})
 
 Reference
-: ([Rankers 1998](#orgce7714c))
+: ([Rankers 1998](#org1e7e43d))
 
 Author(s)
 : Rankers, A. M.
@@ -163,13 +163,13 @@ The basic questions that are addressed in this thesis are:
 
 ### Basic Control Aspects {#basic-control-aspects}
 
-A block diagram representation of a typical servo-system is shown in Figure [1](#org65d822a).
+A block diagram representation of a typical servo-system is shown in Figure [1](#org8d51e97).
 The main task of the system is achieve a desired positional relation between two or more components of the system.
 Therefore, a sensor measures the position which is then compared to the desired value, and the resulting error is used to generate correcting forces.
 In most systems, the "actual output" (e.g. position of end-effector) cannot be measured directly, and the feedback will therefore be based on a "measured output" (e.g. encoder signal at the motor).
 It is important to realize that these two outputs can differ, first due to resilience in the mechanical system, and second because of geometrical imperfections in the mechanical transmission between motor and end-effector.
 
-<a id="org65d822a"></a>
+<a id="org8d51e97"></a>
 
 {{< figure src="/ox-hugo/rankers98_basic_el_mech_servo.png" caption="Figure 1: Basic elements of mechanical servo system" >}}
 
@@ -180,10 +180,10 @@ The correction force \\(F\\) is defined by:
 F = k\_p \epsilon + k\_d \dot{\epsilon} + k\_i \int \epsilon dt
 \end{equation}
 
-It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org63224e1)).
+It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org7b90722)).
 If \\(c\\) and \\(d\\) represent the constant mechanical stiffness and damping between points \\(A\\) and \\(B\\), and a reference position profile \\(h(t)\\) is applied at \\(A\\), then an opposing force \\(F\\) is generated as soon as the position \\(x\\) and speed \\(\dot{x}\\) of point \\(B\\) does not correspond to \\(h(t)\\) and \\(\dot{h}(t)\\).
 
-<a id="org63224e1"></a>
+<a id="org7b90722"></a>
 
 {{< figure src="/ox-hugo/rankers98_basic_elastic_struct.png" caption="Figure 2: Basic Elastic Structure" >}}
 
@@ -199,9 +199,9 @@ These properties are very essential since they introduce the issue of **servo st
 
 An important aspect of a feedback controller is the fact that control forces can only result from an error signal.
 Thus any desired set-point profile first leads to a position error before the corresponding driving forces are generated.
-Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#org5cef574).
+Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#org4adf108).
 
-<a id="org5cef574"></a>
+<a id="org4adf108"></a>
 
 {{< figure src="/ox-hugo/rankers98_feedforward_example.png" caption="Figure 3: Mechanical servo system with feedback and feedforward control" >}}
 
@@ -246,30 +246,161 @@ Basically, machine dynamics can have two deterioration effects in mechanical ser
 
 #### Actuator Flexibility {#actuator-flexibility}
 
-<a id="org2877580"></a>
+The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#org72d5adf)) due to compliance between the motor and the load.
+
+<a id="org72d5adf"></a>
 
 {{< figure src="/ox-hugo/rankers98_actuator_flexibility.png" caption="Figure 4: Actuator Flexibility" >}}
 
 
 #### Guiding System Flexibility {#guiding-system-flexibility}
 
-<a id="orgabbbd68"></a>
+The second category of dynamic phenomena results from the limited stiffness of the guiding system in combination with the fact the the device is driven in such a way that it has to rely on the guiding system to suppress motion in an undesired direction (in case of a linear direct drive system this occurs if the driving force is not applied at the center of gravity).
+
+In general, a rigid actuator possesses six degrees of freedom, five of which need to be suppressed by the guiding system in order to leave one mobile degree of freedom.
+In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#orgc8208b1)).
+
+<a id="orgc8208b1"></a>
 
 {{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}}
 
+The carriage is free to move in the guiding direction \\(x\\), whereas the perpendicular displacement \\(y\\) and the rotation \\(\phi\\) is prevented via two fixtures with limited stiffness \\(c\\).
+The limited support stiffness and the inertia properties of the actuator will result in two resonances, which can be characterized as perpendicular mode and rocking mode.
+
+Every actuator as some sort of guiding system in order to suppress certain DoF, and thus possesses guiding modes.
+However, whether this leads to dynamic problems depends very much on the location of the driving force and the sensor.
+By choosing the proper location of the driving force one can avoid excitation of these modes, whereas the location of the sensor influences the effect of such a mode on the servo stability where excitation of the mode could not be avoided.
+
+In general, it should be attempted to design the actuator (mass distribution and location of driving force) such that it will perform the desired motion even in the absence of the guiding system.
+
 
 #### Limited Mass and Stiffness of Stationary Machine Part {#limited-mass-and-stiffness-of-stationary-machine-part}
 
-<a id="org7d3b694"></a>
+The last category of dynamic phenomena results from the limited mass and stiffness of the stationary part of a mechanical servo-system.
+In contrast to many textbooks on mechanics and machine dynamics, it is good practice always to look at the combination of driving force on the moving part, and reaction force on the stationary part, of a positioning device.
+When doing so, one has to consider what the effect of the reaction force on the systems performance will be.
+In the discussion of the previous two dynamic phenomena, the stationary part of the machine was assumed to be infinitely stiff and heavy, and therefore the effect of the reaction force was negligible.
+However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#org1252ea7)).
+
+<a id="org1252ea7"></a>
 
 {{< figure src="/ox-hugo/rankers98_limited_m_k_stationary_machine_part.png" caption="Figure 6: Limited Mass and Stiffness of Stationary Machine Part" >}}
 
 
-## Modal Decomposition {#modal-decomposition}
+## [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
 
 
 ### Mathematics of Modal Decomposition {#mathematics-of-modal-decomposition}
 
+The general equation of motion of a linear mechanical system with a finite number of DoF, and without damping is:
+
+\begin{equation}
+M \ddot{x}(t) + K x(t) = f(t)
+\end{equation}
+
+in which \\(M\\) and \\(K\\) stand for the symmetric semi-positive definite mass and stiffness matrix, \\(x(t)\\) and \\(\ddot{x}(t)\\) represent the displacement and acceleration vectors, and \\(f(t)\\) denotes the vector of forces.
+
+Generally this system of equations is coupled but it can always be decoupled by using a transformation based on the non-trivial solutions (the eigenvectors) of the following eigenvalue problem:
+
+\begin{equation}
+(K + \omega\_i^2 M) \phi\_i = 0
+\end{equation}
+
+Solving the eigenvalue problem gives the eigenvalues \\(\omega\_1^2, \omega\_2^2, \dots, \omega\_n^2\\) and the corresponding eigenvectors or mode-shape vectors \\(\phi\_1, \phi\_2, \dots, \phi\_n\\).
+
+These eigenvectors have the following orthogonality properties, or can always be chosen such that:
+
+\begin{equation} \label{eq:eigenvector\_orthogonality\_mass}
+\phi\_i^T M \phi\_j = 0 \quad (i \neq j)
+\end{equation}
+
+For \\(i=j\\) the result of the multiplication according to equation \eqref{eq:eigenvector_orthogonality_mass} yields a non-zero result, which is normally indicated as modal mass \\(\mathit{m}\_i\\):
+
+\begin{equation} \label{eq:modal\_mass}
+\phi\_i^T M \phi\_i = \mathit{m}\_i
+\end{equation}
+
+Because only the direction but not the length of an eigenvector is defined, several scaling methods are used, all based on equation \eqref{eq:modal_mass}:
+
+-   \\(|\phi\_i| = 1\\):: Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation \eqref{eq:modal_mass}.
+-   \\(\max(\phi\_i) = 1\\):: Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation \eqref{eq:modal_mass}.
+-   \\(m\_i = 1\\):: The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation \eqref{eq:modal_mass}.
+
+The orthogonality properties also apply to the stiffness matrix \\(K\\):
+
+\begin{align}
+\phi\_i^T K \phi\_j &= 0 \quad (i \neq j) \\\\\\
+\phi\_i^T K \phi\_i &= \omega\_i^2 \mathit{m}\_i = \mathit{k}\_i
+\end{align}
+
+Because the \\(n\\) eigenvectors \\(\phi\_i\\) form a base in the n-dimensional space, any displacement vector \\(x(t)\\) can be written as a linear combination of the eigenvectors.
+Let \\(q\_i(t)\\) be the response of the decopled mode \\(i\\), then the resulting displacement vector \\(x(t)\\) will be:
+
+\begin{equation}
+x(t) = q\_1(t) \phi\_1 + q\_2(t) \phi\_2 + \dots + q\_n(t) \phi\_n
+\end{equation}
+
+For one individual physical DoF \\(x\_k\\):
+
+\begin{equation}
+x(t) = q\_1(t) \phi\_{1k} + q\_2(t) \phi\_{2k} + \dots + q\_n(t) \phi\_{nk}
+\end{equation}
+
+with \\(\phi\_{ik}\\) being the element of the mode-shape vector \\(\phi\_i\\) that corresponds to the physical DoF \\(x\_k\\).
+
+The physical interpretation of the above two equations is that any motion of the system can be regarded as a combination of the contribution of the various modes.
+
+On can combine the eigenvectors in a matrix \\(\Phi\\) and the coefficients \\(q\_i\\) in a vector \\(q(t)\\) which leads to:
+
+\begin{equation}
+x(t) = \Phi q(t)
+\end{equation}
+
+With:
+
+\begin{align\*}
+ \Phi &= \begin{bmatrix} \phi\_1 & \phi\_2 & \dots & \phi\_n \end{bmatrix} \\\\\\
+ q(t) &= \begin{bmatrix} q\_1(t) \\ q\_2(t) \\ \vdots \\ q\_n(t) \end{bmatrix}
+\end{align\*}
+
+Substitution of \\(x(t) = \Phi q(t)\\) into the original equation of motion and premultiplication with \\(\Phi^T\\) results in:
+
+\begin{equation}
+\Phi^T M \Phi \ddot{q}(t) + \Phi^T K \Phi q(t) = \Phi^T f(t)
+\end{equation}
+
+Which finally leads to a set of **uncoupled** equations of motion that describe the contribution of each mode:
+
+\begin{equation}
+\begin{bmatrix}
+m\_1 & & & \\\\\\
+& m\_2 & & \\\\\\
+& & \ddots & \\\\\\
+& & & m\_n
+\end{bmatrix} \begin{bmatrix}
+\ddot{q}\_1 \\ \ddot{q}\_2 \\ \vdots \\ \ddot{q}\_n
+\end{bmatrix} + \begin{bmatrix}
+k\_1 & & & \\\\\\
+& k\_2 & & \\\\\\
+& & \ddots & \\\\\\
+& & & k\_n
+\end{bmatrix} \begin{bmatrix}
+q\_1 \\ q\_2 \\ \vdots \\ q\_n
+\end{bmatrix} = \begin{bmatrix}
+\phi\_1^T f \\ \phi\_2^T f \\ \vdots \\ \phi\_n^T f
+\end{bmatrix}
+\end{equation}
+
+For the i-th modal coordinate \\(q\_i\\) the equation of motion is:
+
+\begin{equation}
+m\_i \ddot{q\_i}(t) + k\_i q\_i(t) = \phi\_i^T f(t)
+\end{equation}
+
+which is a simple second order differential equation similar to that of a single mass spring system.
+Using basic formulae that are derived for a simple mass spring system, one is now able to analyse the time and frequency response of all individual modes.
+Having done that, the total motion of the system can simply be obtained by summing the contributions of all modes.
+
 
 ### Graphical Representation {#graphical-representation}
 
@@ -337,4 +468,4 @@ Through the enormous performance drive in mechatronics systems, much has been le
 
 ## Bibliography {#bibliography}
 
-<a id="orgce7714c"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
+<a id="org1e7e43d"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.