Modal Participation Factors

Mode Shapes

Every structure has the tendency to vibrate at a given set of natural frequencies. Each natural frequency is associated with a shape, called mode shape, that the model tends to assume when vibrating at that frequency.

Modal participation factors

SimSolid calculates modal participation, effective mass, and cumulative mass factors for each mode in a specified global or local coordinate reference frame for only flexible modes. Rigid body modes will be ignored.

Modal participation factors are scalars that measure the interaction between the modes and the directional excitation in a given reference frame. Larger values indicate a stronger contribution to the dynamic response.

Modal participation factor,  γ i = ϕ i T M V MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeytaiaab+ gacaqGKbGaaeyyaiaabYgacaqGGaGaaeiCaiaabggacaqGYbGaaeiD aiaabMgacaqGJbGaaeyAaiaabchacaqGHbGaaeiDaiaabMgacaqGVb GaaeOBaiaabccacaqGMbGaaeyyaiaabogacaqG0bGaae4Baiaabkha caqGSaGaaeiiaiabeo7aNnaaBaaaleaacaWGPbaabeaakiabg2da9m aacmaabaGaeqy1dy2aa0baaSqaaiaadMgaaeaacaWGubaaaaGccaGL 7bGaayzFaaWaamWaaeaacaWGnbaacaGLBbGaayzxaaWaaiWaaeaaca WGwbaacaGL7bGaayzFaaaaaa@5E52@

Where,
V MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WGwbaacaGL7bGaayzFaaaaaa@3903@
Unit translational direction vector
ϕ MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq aHvpGzaiaawUhacaGL9baaaaa@39F0@
Normalized mode shape
[ M ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaad2 eacaGGDbaaaa@3889@
Mass matrix
Modal participation factor ratio

The modal participation factor ratio is the modal participation factor for each translational direction divided by the maximum modal participation factor of all the modes for that direction. So, each of the three directions will have a value of 1.0 for the mode that has the maximum modal participation factor, and the other modes will have a value less than 1.0.

Effective mass factors

The Effective mass factors associated with each mode represent the amount of system mass participating in that mode in a given excitation direction. This value is given as a percentage of the total system mass.

Effective mass = γ i 2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyraiaabA gacaqGMbGaaeyzaiaabogacaqG0bGaaeyAaiaabAhacaqGLbGaaeii aiaab2gacaqGHbGaae4CaiaabohacqGH9aqpcqaHZoWzdaqhaaWcba GaamyAaaqaaiaaikdaaaaaaa@470A@

Effective mass along direction is the square of modal participation factor of that direction. Therefore, a mode with a large effective mass will be a significant contributor to the system’s response in the given excitation direction.

A common rule of thumb for linear dynamic analysis is that a mode should be included if it contributes more than 1-2% of the total effective mass.

Cumulative mass

The cumulative mass for mode n is the sum of the effective mass factors for modes 1 through n. A common rule of thumb for linear dynamic analysis is to include sufficient modes such that the cumulative mass is at least 80% in the predominant direction of excitation vibration.