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Some objects have a natural frequency. This can be anything from a metal ball to a table, etc. When we hit such an object, it will start vibrating with a certain frequency $f$. Because of damping the amplitude of this oscillation will decrease with time.

In continuum mechanics, an object consists of many particles, which are also called material points.

Every particle will also be moving through space in an oscillating way. This means that every particle also has its own frequency. Because the particles are interacting with each other, I suspect that the frequencies of neighbouring particles should be very close to each other.

In a natural vibration, are the vibrations of all particles exactly equal to each other? Or are they only approximately equal, and does it vary throughout the material? Can one prove this from general principles? Or is natural frequency defined in this way?

Formulas

To clarify my question, here are some formulas.

Let $\vec{r}_n(t)$ denote the position of particle $n$ at time $t$. If $N$ is the number of particles and all particles have mass $m$, then the center of mass is at $\vec{CM}(t)=\frac1N\sum_n \vec{r}_n(t)$.

The entire object is vibrating at frequency $f$. This means that (at least approximately) $\vec{CM}(t)$ is a periodic function of time with period $\frac{1}{f}$.

Every particle in the object is also following a trajectory that is periodic in time. Therefore we can calculate the period and thus the frequency $f_i$ for every single particle. Now the question is, whether $f_i=f$ for all $i$, or whether there may be small differences in frequency for different $i$.

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  • $\begingroup$ do you wish a description at macroscopic level, assuning the hypothesis of continuum holds? If so, I'll give you an answer. BTW, a continuum doesn't have only one natural frequency, but in principle an infinite number of natural frequencies, each one with its proper natural mode meant as the "shape of the vibration" at that given frequency. $\endgroup$
    – basics
    Feb 2 at 13:29
  • $\begingroup$ Then, always focusing on linear structures, you need to think at the response of the system as the combination of free and response to an external force. Given a small damping, free response decays for stable structures, and only forced response remains: frequency response tells us that the response of the system shows the same frequencies contained in the forcing term, with amplitude and phase depending on the structure $\endgroup$
    – basics
    Feb 2 at 13:34
  • $\begingroup$ @basics Yes, i would assume the continuum hypothesis. Also, I am interested in the free response. Although it decays, it should still have a natural frequency in the time interval that it exists $\endgroup$
    – Riemann
    Feb 2 at 13:42

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In this answer, I'll first focus on non-damped systems, and later introduce damping with the modal damping assumption, common in structural mechanics.

In this answer, let's assume:

  • we're dealing with an elastic medium, in the regime of small displacements and strain, governed by the differential problem \begin{equation}\begin{cases} & - \nabla \cdot \mathbb{T} = \boldsymbol{f} \ , & & \qquad \boldsymbol{r} \in \Omega \\ & \boldsymbol{u} = \boldsymbol{u}_b \ , & & \qquad \boldsymbol{r} \in \Gamma_D \\ & \boldsymbol{\hat{n}} \cdot \mathbb{T} = \boldsymbol{t}_n \ , & & \qquad \boldsymbol{r} \in \Gamma_N \ \end{cases}\end{equation}

  • it's possible to write the displacement field of each point of the medium $\mathbf{u}(\mathbf{r},t)$ with a finite dimensional approximation $$ \mathbf{u}(\mathbf{r},t) = \sum_{k=1}^{n} \boldsymbol{\phi}_k(\mathbf{r}) u_k(t) = \left[ \boldsymbol{\phi}_1(\mathbf{r}), \boldsymbol{\phi}_2(\mathbf{r}), \dots \boldsymbol{\phi}_{n}(\mathbf{r}) \right] \begin{bmatrix} u_1(t) \\ \dots \\ u_{n}(t) \end{bmatrix} = \underline{\underline{\Phi}}(\mathbf{r}) \underline{u}(t) $$ so that the dynamical equations of the non-damped systems can be recast as $$ \underline{\underline{M}} \underline{\ddot{u}} + \underline{\underline{K}} \underline{u} = \underline{f} \ , $$ with $\underline{\underline{M}}$, $\underline{\underline{K}}$ the mass and the stiffness matrices of this system;

  • structural damping can be described by modal damping, (see below)

1.1 - Dynamical equations and modal structure of the system: on-damped system

Set of orthogonal eigenvectors with pure imaginary eigenvalues \begin{equation} j \omega_k, \ \underline{\hat{u}}_k \end{equation}

Using the modal basis $\underline{\underline{U}}$ to express the "nodal variables" $\underline{u}$ into "modal amplitudes" $\underline{q}$ (it's the projection of "nodal amplitudes" onto "modal amplitudes"), $$ \underline{u}(t) = \underline{\underline{U}} \underline{q}(t) \tag{1} $$ Exploiting the orthogonality of the eigenvectors w.r.t. both mass and stiffness matrices (prove it!), the diagonalization of the system $$ \underline{\underline{U}}^T \left( \underline{\underline{M}} \underline{\underline{U}} \ddot{\underline{q}}(t) + \underline{\underline{M}} \underline{\underline{U}} \underline{q}(t) \right) = \underline{\underline{U}}^T \underline{f} \ , $$ gives $n$ independent equations (or a diagonal system) for the modal amplitudes $$ m_k \ddot{q}_k(t) + k_k q_k(t) = U_{ik} f_i(t) \ , $$ where the modal forcing is the projection of the "nodal force" $\underline{f}$ over the mode $\underline{\hat{u}}_k$.

Remark. The amplitude of each natural mode is governed by the equations of a simple harmonic oscillator forced by the projection of the nodal force on that mode. The nodal amplitude is the linear combination of the modes, modulated by their amplitudes (each one with its own natural frequency and modal damping), through the expansion (1).

1.2 - Dynamical equations and modal structure of the system: modal damping

Once the modal base is evaluated, it's possible to introduce modal damping $c_k$ in the modal equations, $$ m_k \ddot{q}_k(t) + c_k \dot{q}_k(t) + k_k q_k(t) = U_{ik} f_i(t) \ , $$ or making the natural frequency $\omega^2_{n,k} = \frac{k_k}{m_k}$ and damping coefficient explicit $\xi_k$, $$ \ddot{q}_k(t) + 2 \omega_{n,k} \xi_{k} \dot{q}_k(t) + \omega_k^2 q_k(t) = \frac{U_{ik}}{m_k} f_i(t) \ . \tag{2} $$

  1. Response

Now, it's time to evaluate the response of the system to specific forcing. 2.1 - Free response to initial conditions (~ impulsive force)

Given initial conditions on the "nodal displacement", $$\begin{cases} \underline{u}(0) = \underline{u}_0 \\ \underline{\dot{u}}(0) = \underline{v}_0 \end{cases}$$ the corresponding amplitudes of the modal variables read $$\begin{cases} \underline{q}(0) = \underline{\underline{U}}^T \underline{u}_0 \\ \underline{\dot{q}}(0) = \underline{\underline{U}}^T \underline{v}_0 \end{cases} \qquad \text{i.e.} \qquad \begin{cases} q_k(0) = \underline{\hat{u}}_k^T \underline{u}_0 \\ \dot{q}_k(0) = \underline{\hat{u}}_k^T \underline{v}_0 \\ \end{cases} $$ This initial conditions can be used to solve for the modal variables $q_k(t)$ using modal equations (2), and retrieving the nodal response from (1).

2.2 - Harmonic response (~ response to harmonic force after initial transient is over)

The very same procedure in time domain can be followed for a response to harmonic forcing.

This case can be also studied in Fourier domain, where the modal dynamical equations (2) become $$ \left[ -\omega^2 + j 2 \xi_k \omega_{n,k} \omega + \omega_{n,k}^2 \right] \hat{q}_k(\omega) = \dfrac{\underline{\hat{u}}_k^T }{m_k}\underline{\hat{f}}(\omega) \ , $$ so that it's easy to relate the forcing to the modal displacement through a transfer function $$ \hat{q}_k(\omega) = \left[ -\omega^2 + j 2 \xi_k \omega_{n,k} \omega + \omega_{n,k}^2 \right]^{-1} \dfrac{\underline{\hat{u}}_k^T }{m_k}\underline{\hat{f}}(\omega) $$

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  • $\begingroup$ So then for each given mode, the frequency is global, and can be found via the corresponding eigenvalue $\endgroup$
    – Riemann
    Feb 2 at 17:25
  • $\begingroup$ yes. In general, each mode is global, i.e. it is non null almost everywhere (except for what is called a knots). However, in some structure some "local" modes may exists, modes that are approximately zero in a large region of the structure. Let's take as example the frame of an airplane: usually some global modes are the bending and torsion of the wings, more or less coupled with motion of the fuselage; but it's likely that some "local mode" exists, as an example, associated with the bending of a single panel of the skin $\endgroup$
    – basics
    Feb 2 at 17:45

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