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# 物理代写|振动力学代写Vibration Mechanics代考|ENG211 Beautiful Features of Vibration Mechanics

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## 物理代写|振动力学代写Vibration Mechanics代考|Beautiful Features of Vibration Mechanics

In the theoretical frame of vibration mechanics, unity exists between different parts of the frame, between parts and the whole frame as well, such that the frame is a unified and beautiful entirety. For instance, the dynamic equations of a discrete system are a set of ordinary differential equations of the second-order of time derivative, while the dynamic equation of a continuous system is a partial differential equation of the second-order of time derivative. If a discrete system is divided into several sub-systems, the dynamic equations of each sub-system keep as a set of ordinary differential equations similar to those of the entire system. Thus, the theory of vibration mechanics enjoys a quite unified frame. Once a breakthrough is made, it will bring the development of the whole field.

The elementary textbooks mainly dealt with the linear vibration systems in a discrete form. That is, the linear dynamic system with an input vector $\boldsymbol{f}(t)$ and an output vector $\boldsymbol{u}(t)$. The linear system has two fundamental properties, namely, homogeneity and additivity. Mathematically speaking, the system can be defined as a linear mapping $L: f(t) \mapsto u(t)$ from an input vector to an output vector, while these two properties can be expressed as
$$\left{\begin{array}{l} L[\alpha \boldsymbol{f}(t)]=\alpha L[\boldsymbol{f}(t)]=\alpha \boldsymbol{u}(t), \quad \alpha \in(-\infty,+\infty) \ L\left[\boldsymbol{f}_1(t)+\boldsymbol{f}_2(t)\right]=L\left[\boldsymbol{f}_1(t)\right]+L\left[\boldsymbol{f}_2(t)\right]=\boldsymbol{u}_1(t)+\boldsymbol{u}_2(t) \end{array}\right.$$
In the community of mechanics, the two properties are named the principle of superposition. That is, when the multiple inputs, including initial disturbance and multiple external loads, are applied to a linear system, the entire output, namely, the dynamic response, is the superposition of all outputs, each of which is caused by an individual input. The theory of linear vibrations based on the principle of superposition exhibits the beautiful unity.

To be more specific, consider a linear $n$-DoF system, which yields the following initial value problem governed by a set of linear ordinary differential equations
$$\left{\begin{array}{l} M \ddot{u}(t)+\boldsymbol{C} \dot{u}(t)+\boldsymbol{K} \boldsymbol{u}(t)=\boldsymbol{f}(t) \ \boldsymbol{u}(0)=u_0, \quad \dot{\boldsymbol{u}}(0)=\dot{u}_0 \end{array}\right.$$
where $f: t \mapsto \mathbb{R}^n$ is the external force vector, $u: t \mapsto \mathbb{R}^n$ is the system displacement vector, $\boldsymbol{u}_0 \in \mathbb{R}^n$ is the initial displacement vector, $\dot{u}_0 \in \mathbb{R}^n$ is the initial velocity vector, $\mathbb{R}^n$ is a real vector space of $n$ dimensions and named the configuration space of the system. In addition, $M \in \mathbb{R}^{n \times n}$ is the positive definite mass matrix, $\boldsymbol{K} \in \mathbb{R}^{n \times n}$ is the positive or semi positive definite stiffness matrix, $C \in \mathbb{R}^{n \times n}$ is the symmetric damping matrix, and $\mathbb{R}^{n \times n}$ is the space of real square matrix of order $n$.

## 物理代写|振动力学代写Vibration Mechanics代考|Simplicity

The theoretical frame of vibration mechanics has a clear outlines, the minimal concepts of component elements, and a very simple structure. For instance, the theoretical frame of linear vibrations is based on the principle of superposition and gives rise to a few simple, but significant theorems and formulae. Among them, Eq. (2.2.3) is a good example.

In addition, the vibration problems often have the possibility of decomposition. As stated in Sect. 2.2.1, for instance, the response of a linear system includes two parts. One is the system response subject to the initial disturbance, and the other is the response subject to the external excitation.

The other example is the complicated time history of quasi-periodic vibration $w(t)$ shown in Fig. 2.1a, which can be decomposed, via Fourier transform, into three harmonic vibrations in frequency domain. Figure $2.1 \mathrm{~b}$ presents the simple picture of the amplitude-frequency relation $|W(f)|$.

The main thread in the study of vibration problems is to decompose a complicated problem into a number of simpler ones. The main thread has led to a number of well known methods, such as the modal analysis, the Duhamel integral in time domain, the spectral analysis in frequency domain, and the dynamic sub-structuring in space domain. Now, we turn to the modal analysis of a linear MDoF system to explain the main thread as follows.

Let $u_r(t)=\varphi_r \sin \left(\omega_r t+\theta_r\right), \quad r=1,2, \ldots, n$ be the natural vibrations of the linear system governed by Eq. (2.2.2). They satisfy the following solutions of an eigenvalue problem
$$\left(\boldsymbol{K}-\omega_r^2 \boldsymbol{M}\right) \boldsymbol{\varphi}_r=\mathbf{0}, \quad r=1,2, \ldots, n$$
If all the natural frequencies are distinct, namely, $0 \leq \omega_1<\omega_2<\cdots<\omega_n$, the mode shape vectors $\varphi_r \in \mathbb{R}^n, \quad r=1,2, \ldots, n$ satisfy the orthogonal relations, or called the orthogonality, as follows

$$\boldsymbol{\varphi}_r^{\mathrm{T}} \boldsymbol{M} \boldsymbol{\varphi}_s=\left{\begin{array}{l} M_r, r=s \ 0, \quad r \neq s ; \end{array} \quad \boldsymbol{\varphi}_r^{\mathrm{T}} \boldsymbol{K} \boldsymbol{\varphi}_s=\left{\begin{array}{l} K_r, r=s \ 0, r \neq s ; \end{array} \quad r, s=1,2, \ldots, n\right.\right.$$
where $M_r>0$ is the $r$-th modal mass coefficient, and $K_r=M_r \omega_r^2 \geq 0$ is the $r$-th modal stiffness coefficient.

## 物理代写|振动力学代写振动力学代考|振动力学的美丽特征

$$\left{\begin{array}{l} L[\alpha \boldsymbol{f}(t)]=\alpha L[\boldsymbol{f}(t)]=\alpha \boldsymbol{u}(t), \quad \alpha \in(-\infty,+\infty) \ L\left[\boldsymbol{f}_1(t)+\boldsymbol{f}_2(t)\right]=L\left[\boldsymbol{f}_1(t)\right]+L\left[\boldsymbol{f}_2(t)\right]=\boldsymbol{u}_1(t)+\boldsymbol{u}_2(t) \end{array}\right.$$

$$\left{\begin{array}{l} M \ddot{u}(t)+\boldsymbol{C} \dot{u}(t)+\boldsymbol{K} \boldsymbol{u}(t)=\boldsymbol{f}(t) \ \boldsymbol{u}(0)=u_0, \quad \dot{\boldsymbol{u}}(0)=\dot{u}_0 \end{array}\right.$$

## 物理代写|振动力学代写振动力学代考|简单

$$\left(\boldsymbol{K}-\omega_r^2 \boldsymbol{M}\right) \boldsymbol{\varphi}_r=\mathbf{0}, \quad r=1,2, \ldots, n$$

$$\boldsymbol{\varphi}_r^{\mathrm{T}} \boldsymbol{M} \boldsymbol{\varphi}_s=\left{\begin{array}{l} M_r, r=s \ 0, \quad r \neq s ; \end{array} \quad \boldsymbol{\varphi}_r^{\mathrm{T}} \boldsymbol{K} \boldsymbol{\varphi}_s=\left{\begin{array}{l} K_r, r=s \ 0, r \neq s ; \end{array} \quad r, s=1,2, \ldots, n\right.\right.$$

## Matlab代写

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