# Dynamic Analysis of the Musical Triangles--Experimental and Numerical Approaches
## 2 Materials and Methods
### Methods
#### 2.2.3 Modal Transient Response Analysis  
Transient dynamic response analysis is the most general method of calculating the forced dynamic response. The purpose of such an analysis is to determine the behavior of a structure subjected to an excitation that varies over time [20; 21; 22; 23; 24; 25]. Transient excitation is explicitly defined in the time domain. All loadings applied to the structure are known at all times, and the results of a transient response analysis are usually displacements, velocities, and accelerations of the points in the model and stresses in the elements. The dynamic analysis in the transient mode was also performed with the Simcenter 12 software; for this type of analysis the SOL 112 Modal Transient Response solution was used [21]. The principle of the method consists of transforming the physical coordinates denoted by \(\{u(t)\}\) into modal coordinates denoted by \(\{\xi(t)\}\), assuming that  
\[\{u(t)\}=[\phi]\{\xi(t)\}, \tag{1}\]  
where \([\phi]\) is the mode shapes. Taking into account the equation of motion \(\{P(t)\}\) (2) and ignoring temporarily damping, Equation (3) is obtained [21; 22; 23; 24; 25]:  
\[[M]\big{\{}\ddot{u}(t)\big{\}}+[B]\big{\{}\dot{u}(t)\big{\}}+[K]\big{\{}u(t) \big{\}}=\{P(t)\}, \tag{2}\]  
\[[M]\big{\{}\ddot{u}(t)\big{\}}+[K]\big{\{}u(t)\big{\}}=\{P(t)\}, \tag{3}\]  
where \([M]\), \([B]\), and \([K]\) are the matrices of mass, viscous damping, and stiffness, respectively. Replacing in (3) relation (1), results coupled equation of motion (4):  
\[[M][\phi]\Big{\{}\ddot{\xi}(t)\Big{\}}+[K][\phi]\{\xi(t)\}=\{P(t)\}, \tag{4}\]  
To uncouple the equation of motion, we multiply Equation (4) by \([\phi]^{T}\) to obtain (5):  
\[[\phi]^{T}[M][\phi]\Big{\{}\ddot{\xi}\Big{\}}+[\phi]^{T}[K][\phi]\{\xi\}=[ \phi]^{T}\{P(t)\}, \tag{5}\]  
where \([\phi]^{T}[M][\phi]\) is modal generalized mass matrix; \([\phi]^{T}[K][\phi]\)--modal generalized stiffness matrix; \([\phi]^{T}\{P(t)\}\)--modal force vector. Since the matrices of mass and modal stiffness are diagonal matrices and have no off-diagonal terms, the equations of motion are uncoupled and can be written as a set of systems with a single degree of freedom (6) [21; 22; 23; 24; 25]:  
\[m_{i}\ddot{\xi}_{i}(t)+k_{i}\xi_{i}(t)=p_{i}(t), \tag{6}\]  
where \(m_{i}\) is the modal mass for the \(i\)th mode; \(k_{i}\)--modal stiffness for the \(i\)th mode; \(p_{i}\)--modal force for the \(i\)th mode. Once the individual modal responses, \(\xi_{i}(t)\), are computed, physical responses are recovered as the summation of the modal responses using (7):  
\[\{u(t)\}=[\phi]\{\xi(t)\}, \tag{7}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Triangle** & \
\begin{tabular}{c} **Elasticity** \\ **Modulus** \\ **(MPa)** \\ \end{tabular} & **Density (kg/m\({}^{3}\))** &
\begin{tabular}{c} **Poisson** \\ **Coefficient** \\ \end{tabular} & **Materials** \\ \hline T1 & 184,479 & 7818 & 0.30 & Stainless steel \\ T2 & 68,382 & 2708 & 0.33 & Aluminum \\ T3 & 68,382 & 2708 & 0.33 & Aluminum \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The engineering constants of triangle materials. In this study, the experimentally determined structural damping was introduced in the transient analysis, taking into account the dominant frequency at which the damping is active [21; 22; 23; 24; 25].