The phrase "and the second modes" is correct and usable in written English.
It can be used when referring to multiple modes or methods in a discussion or explanation, particularly when comparing or listing them.
Example: "In our analysis, we will consider the first mode of operation and the second modes to determine their effectiveness."
Alternatives: "as well as the second modes" or "along with the second modes".
The obtained eigenvectors of the first and the second modes indicated the Kármán's-type structure of vortex shedding and the result of their reconstruction was able to highlight the dominant structure of prism wake interaction in the vortex-induced oscillation.
Moreover, the safety operation range is defined as the interval between the natural frequencies for the first and the second modes, and finally, the purpose is to raise the natural frequency for second mode for a given constraint conditions changing the design parameters.
We also observed that omission of the transverse shear effect, particularly at the ply angles between 60° and 120°, led to an overestimation of the natural frequencies of the first and the second modes for higher slenderness ratios, while overlapping results are obtained for any thickness or aspect ratios regardless of including/omission of transverse shear deformation.
As illustrated in Figure 3, we utilize only the first M2 highest eigenvalues of the first and of the second modes to estimate the model order.
The first growth mode, frequently employed by engineers, is characterised by direct bending strains through the thickness, and the second mode, mainly apparent in biological systems, is driven by extensional strains of the middle surface.
Correspondingly, in the case of bimodal distribution, the following rule was applied: The first mode in the distribution is within dbh classes from 10 − 20 cm, and the second mode in the distribution is after the 25 30-cm 25 30-cms.
This corroborates the benefits provided by forward-backward-averaging that it decorrelates coherent sources and contributes to an enhanced accuracy of the parameter estimation.In Figures 7 and 8, the corresponding spatial frequency estimates of the first mode and the second mode averaged over 1,000 trials are illustrated, respectively.
By considering the first vibration mode in addition to the two asymmetric vibration modes, quadratic nonlinear terms expressing the interaction between the asymmetric and the first modes appear in the differential equations.
We conducted nodal line optimization for the second and the fifth modes together at the same time, and the results showed that the nodal lines obtained match well with the target nodal lines.
When the rotational component is considered the effects of the first and the third modes which relate to H θ in Eq. 14a, 14b would appear.
H y is related to the first and the third modes in all three cases in Fig. 10b, but the effect of the third mode increases by increasing the effect of the rotational component.
