# Analysis of the Mode Shapes of Kaplan Runners
## 2 Numerical and Experimental Modal Analysis  
The relation between the excitation and the response of the structure is called Frequency Response Function (FRF).  
According to the modal analysis theory, every structure can be modelled as a combination of masses, dampers and springs, which are interconnected with each other. The equation of motion of a system with n degrees of freedom, which is excited with a time-dependent force, is given by:  
\[[M]\ddot{x}+[C]\dot{x}+[K]x=F\;,\;\text{with}\;x=\begin{cases}x_{1}(t)\\ x_{2}(t)\\ x_{3}(t)\\ \vdots\\ x_{n}(t)\end{cases}\text{and}\;F=\begin{cases}f_{1}(t)\\ f_{2}(t)\\ f_{3}(t)\\ \vdots\\ f_{n}(t)\end{cases} \tag{1}\]  
where \(x\), \(\dot{x}\) and \(\ddot{x}\) are displacement, velocity and acceleration vectors, respectively, in the time domain. \([M]\), \([C]\) and \([K]\) are the mass, damping and stiffness matrix of the system. F is the excitation force applied to each degree of freedom.  
To obtain the FRF, Equation (1) has to be transformed from the time domain to the frequency domain. Thus, the Fourier transform is applied to this equation as follows:  
\[\{X(j\omega)\}=[H(j\omega)]\cdot\{F(j\omega)\} \tag{2}\]  
where \(\{X(j\omega)\}\) and \(\{F(j\omega)\}\) are the above time-dependent displacement and force vectors, respectively, in the frequency domain. \([H(j\omega)]\) is the frequency response function, which relates each element of the excitation vector to each element of the displacement vector.  
When a FEM modal analysis is carried out, Equation (1) is often solved as an undamped system, which means \([C]=0\). Natural frequencies and mode shapes are calculated from this model. Then, an individual modal damping factor \(\xi\), may be added to each vibration mode to later calculate the damped natural frequencies. The mode shapes do not change by adding this damping factor. Then, the mode shapes of the undamped system are used in this damped system model [23].  
Many real structures are considered proportionally damped structures. The proportional damping model defines the damping coefficient matrix as a linear combination of the mass and the stiffness matrixes. The FRF for this kind of damped structures is given by [24]:  
\[[H(j\omega)]=\sum_{r=1}^{N}\frac{j2\omega_{r}Q_{r}\{\theta\}_{r}\{\theta\}_{r} ^{t}}{(\theta_{r}^{2}+\omega_{r}^{2}+\omega^{2})-2\theta_{r}j\omega} \tag{3}\]  
In Equation (3), \(N\) is the number of vibration modes of the system. Then, this equation shows a superposition of all vibration mode shapes excited by an external force. Then, for each mode _r_:  
* \(\omega_{r}\) is the natural frequency of the corresponding vibration mode.
* \(\theta_{r}\) is the damping factor of the structure.
* \(\{\theta\}_{r}\) is the vibration mode shape that dominates the structure near a resonance condition (\(\omega_{excitation}=\omega_{r}\)).
* \(Q_{r}\) is the scaling factor, which is a constant value for each mode shape.  
Even though in Equation (2) the excitation force is related to the displacement, the FRF can be represented as:  
* Receptance function: relation between the displacement and the excitation force.
* Mobility function: relation between the velocity and the excitation force.
* Inertance function: relation between the acceleration and the excitation force.  
These three transfer functions are related to each other, as is shown in Equation (4). These transfer functions are equally valid forms of the frequency response function.  
\[[H(j\omega)]=\frac{[H(j\omega)]^{\prime}}{j\omega}=\frac{[H(j\omega)]^{ \prime\prime}}{-\omega^{2}} \tag{4}\]