{"id":28943,"date":"2026-04-23T12:16:52","date_gmt":"2026-04-23T12:16:52","guid":{"rendered":"https:\/\/support.buildsoft.eu\/?post_type=ht_kb&p=28943"},"modified":"2026-04-23T12:17:15","modified_gmt":"2026-04-23T12:17:15","slug":"dld-02-mass-spring-system-with-2-degrees-of-freedom-modal-information","status":"publish","type":"ht_kb","link":"https:\/\/support.buildsoft.eu\/es\/knowledge-base\/dld-02-mass-spring-system-with-2-degrees-of-freedom-modal-information\/","title":{"rendered":"DLD 02: Mass-spring system with 2 degrees of freedom \u2013 modal information"},"content":{"rendered":"

Description<\/h2>\n

\"\"<\/p>\n

\n\n\n\n\n\n\n
Geometry<\/th>\nMass<\/td>\n\"m_1=50kg\"
\n\"m_2=80kg\"<\/td>\n<\/tr>\n
Spring contant<\/td>\n\"k_1=4000N/m\"
\n\"k_2=6000N/m\"<\/td>\n<\/tr>\n
Damping ratio’s<\/td>\n\"\xi=0.02\"
\n\"\xi=0.05\"<\/td>\n<\/tr>\n
Mesh<\/th>\nNo. of divisions<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

The damping ratio’s are imposed in the dialog for modal analysis \"\".<\/p>\n

\"\"<\/p>\n

Independent reference results<\/h2>\n
<\/span>Open hand calculations<\/div>
\n

In this example we want to recalculate the modal information. To do so, we’ll be using the same model as in DLD 01.<\/a>
\nThe modal information can be found after this button \"\". Make sure to check the option \u00abDisplay all the information<\/em>\u00ab.<\/p>\n

\"\"<\/p>\n

Natural frequencies \"f\"<\/h4>\n

These are the specific rates at which a system naturally vibrates, determined solely by its mass and stiffness distribution. This data was already available in the validation example DLD 01.<\/a><\/p>\n

  <\/span>   <\/span>\"\[f=\begin{bmatrix}<\/p>\n

Angular frequencies \"\omega\"<\/h4>\n

The relation between the angular frequencies and the natural frequencies is given by:<\/p>\n

  <\/span>   <\/span>\"\[\omega = 2 \pi f=2 \pi \begin{bmatrix}<\/p>\n

Natural periods \"T\"<\/h4>\n

The relation between the natural period and the natural frequency is given by:<\/p>\n

  <\/span>   <\/span>\"\[T= 1/ f= \begin{bmatrix}<\/p>\n

Modal mass \"M\"<\/h4>\n

The shape modes were calculated in DLD 01<\/a>. The first column represents the displacements of mode 1, and the second of mode 2.<\/p>\n

  <\/span>   <\/span>\"\[\phi=\begin{bmatrix}<\/p>\n

The modal mass \"M\" is calculated as:<\/p>\n

  <\/span>   <\/span>\"\[M=\phi^{T} \cdot M_m \cdot \phi=\begin{bmatrix}<\/p>\n

\"M_m\" is the mass matrix \"M_m=\begin{bmatrix}.<\/p>\n

In the Diamonds results for the modal mass \"M\", the \"10^6 kg\" is left out. You just see \u00ab1\u00bb in the table. The mass matrix \"M_m\" cannot be consulted in Diamonds.<\/p>\n

Modal stiffness \"K\"<\/h4>\n

The modal stiffness \"K\" can be calculated in two ways resulting in the same result:<\/p>\n

  <\/span>   <\/span>\"\[K=M \cdot \omega^2=\begin{bmatrix}<\/p>\n

or<\/p>\n

  <\/span>   <\/span>\"\[K=\phi^T \cdot K_k \cdot \phi=\begin{bmatrix}<\/p>\n

\"K_k\" is the stiffness matrix \"K_k=\begin{bmatrix}.<\/p>\n

The stiffness matrix \"K_k\" cannot be consulted in Diamonds.<\/p>\n

Modal damping \"D\"<\/h4>\n

Rayleigh damping was used in this model. In order to calculate the modal damping \"D\", we should calculate the damping matrix \"D_d\" first.<\/p>\n

  <\/span>   <\/span>\"\[a_0=\frac{2 \cdot \omega_1 \cdot \omega_2 \cdot \left( \xi_2 \cdot \omega_1 - \xi_1 \cdot\omega_2 \right)}{\omega_1^2-\omega_1^2}=0.05 \frac{1}{s}\]\"<\/p>\n

  <\/span>   <\/span>\"\[a_1=\frac{2 \cdot \left( \xi_1 \cdot \omega_1 - \xi_2 \cdot\omega_2 \right)}{\omega_1^2-\omega_1^2}=6.11 \cdot 10^{-3} \text{s}\]\"<\/p>\n

The damping matrix \"D_d\" becomes:<\/p>\n

  <\/span>   <\/span>\"\[D_d=a_0\cdot M_m + a_1 \cdot K_k=\begin{bmatrix}<\/p>\n

The damping matrix \"D_d\" cannot be consulted in Diamonds. But using the damping matrix \"D_d\" we can calculate the modal damping \"D\":<\/p>\n

  <\/span>   <\/span>\"\[D=\phi^T \cdot D_d \cdot \phi=\begin{bmatrix}<\/p>\n

Critical damping \"D_c\"<\/h4>\n

Again using the damping matrix \"D_d\" we can caluculate the critical damping \"D_c\":<\/p>\n

  <\/span>   <\/span>\"\[D_c=2 \cdot \sqrt{M \cdot K} = \begin{bmatrix}<\/p>\n

Damping ratio’s \"\xi\"<\/h4>\n

Since this model only has 2 DOF’s, there are only 2 damping ratio’s. If your model contains more DOF, the damping ratio’s of mode 3 until i can be calculated using:<\/p>\n

  <\/span>   <\/span>\"\[\xi_i=\frac{1}{2}\left( \frac{a_0}{2\pi f_i} +a_1 \cdot 2\pi f_i\right)\]\"<\/p>\n

Ratio of the effective modal mass in a mode to the total mass<\/h4>\n

To calculate this ratio we need the modal participation factor \"\gamma\" . The modal participation factor \"\gamma\" cannot be consulted in Diamonds.<\/p>\n

  <\/span>   <\/span>\"\[\gamma=\phi^T \cdot M_m \cdot \begin{bmatrix}<\/p>\n

From the modal participation factor \"\gamma\" we can determine the effective modal mass \"m_eff\". The effective modal mass \"m_eff\" cannot be consulted in Diamonds.<\/p>\n

  <\/span>   <\/span>\"\[m_{eff}=\frac{\gamma^2}{1 \text{kg}}=\begin{bmatrix}<\/p>\n

Note that the sum of 125.94 + 3 .93 = 129.87kg which equals the sum of the masses in the system 50+80kg = 130kg.<\/p>\n

Now let’s calculate the ratio of the effective modal mass to the total mass for each mode. It’s this ratio that you see in Diamonds.<\/p>\n

\"\frac{m_{eff}}{130kg}=\begin{bmatrix} %
\n<\/div><\/div><\/div>\n

Results<\/h2>\n
\n\n\n\n\n\n\n\n\n\n\n\n
<\/th>\nIndependent reference<\/th>\nDiamonds<\/th>\nDifference1<\/a><\/sup><\/th>\n<\/tr>\n
\"f\" [Hz]<\/td>\n0.778
\n2.522<\/td>\n		0.778
\n2.522<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"\omega\" [1\/s]<\/td>\n4.888
\n15.845<\/td>\n		4.886
\n15.837<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"T\" [s]<\/td>\n1.285
\n0.397<\/td>\n		1.286
\n0.397<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"M\"<\/td>\n1
\n1<\/td>\n		1
\n1<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"K\"<\/td>\n23.870
\n250.850<\/td>\n		23.869
\n250.812<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"D\"<\/td>\n0.195
\n1.583<\/td>\n		0.195
\n1.584<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"D_c\"<\/td>\n9.776
\n31.660<\/td>\n		9.771
\n31.674<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n
\"\frac{m_{eff}}{m_{tot}}\"<\/td>\n96.88%
\n3.02%<\/td>\n		96.97%
\n3.03%<\/td>\n		\u22480.00%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

References<\/h2>\n
\n
    \n
  • Hofman, G. E. (2012). Eindige Elementen Methode: Deel 2<\/i>. MERc, p118<\/li>\n<\/ul>\n
      \n
    • Tested in Diamonds 2026.<\/li>\n<\/ul>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

      Description The damping ratio’s are imposed in the dialog for modal analysis . Independent reference results Results References Hofman, G. E. (2012). Eindige Elementen Methode: Deel 2. MERc, p118<\/p>\n","protected":false},"author":3,"comment_status":"closed","ping_status":"closed","template":"","format":"standard","meta":[],"ht-kb-category":[1350],"ht-kb-tag":[],"_links":{"self":[{"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/ht-kb\/28943"}],"collection":[{"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/ht-kb"}],"about":[{"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/types\/ht_kb"}],"author":[{"embeddable":true,"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/comments?post=28943"}],"version-history":[{"count":3,"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/ht-kb\/28943\/revisions"}],"predecessor-version":[{"id":28978,"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/ht-kb\/28943\/revisions\/28978"}],"wp:attachment":[{"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/media?parent=28943"}],"wp:term":[{"taxonomy":"ht_kb_category","embeddable":true,"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/ht-kb-category?post=28943"},{"taxonomy":"ht_kb_tag","embeddable":true,"href":"https:\/\/support.buildsoft.eu\/es\/wp-json\/wp\/v2\/ht-kb-tag?post=28943"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}