Posted on Categories:Vibration Mechanics, 振动力学, 物理代写

# 物理代写|振动力学代写Vibration Mechanics代考|ENME361 Problems of Concern

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## 物理代写|振动力学代写Vibration Mechanics代考|Problems of Concern

(1) Problem 2A: Abnormal effects of damping coefficients
In the hydro-elastic mounting, the oil passes through a set of orifices and produces damping. Hence, it is possible to control damping effect by adjusting the number of orifices and the radii of orifices. Yet, both computations and experiments showed that the resonance suppression seemed almost unchanged when adjusting these parameters. Therefore, engineers felt strange why the hydro-damping did not produce expected results and contradicted to their intuition of mechanics.
(2) Problem 2B: How to determine the degree of freedom of a system In classical dynamics, the degree of freedom, or the DoF for short, of a system is the minimal number of independent generalized coordinates of the system. Then, the degree of freedom of a lumped parameter system is the number of lumped inertial components, such as a lumped mass. The vibration system in Fig. 1.4b has a single lumped mass and seems to be a system of single degree of freedom, or $S D o F$ for short. Yet, it is not possible to use only the vertical displacement $u_1$ of the lumped mass to describe all motions of the system. That is, it is necessary to use the other two vertical displacements $u_2$ and $u_3$ in Fig. 1.4b to describe the motions of two connection points of dashpots and springs. As such, the system seems to have three degrees of freedom, or 3-DoFs for short. Engineers wondered how many degrees of freedom the system possesses.

## 物理代写|振动力学代写Vibration Mechanics代考|Basic Ideas of Study

In fact, the two problems come from the same origin. In elementary textbooks, most vibration systems do not have a spring and a dashpot in serial shown in Fig. 1.4b, where the two connection points do not have any lumped mass so that two degrees of freedom of the system degenerate. The contribution of the dashpot to the system damping ratio of the system, thus, changes.

To understand the above issue, consider a dynamic serial system shown in Fig. 1.5, where a dashpot and a spring in serial constitute Maxwell’s fluidic component ${ }^{4,5}$. Apart from the displacement $u_1(t)$ of the mass block, one needs to use displacement $u_2(t)$ to describe the motion of the connection point between the dashpot and the spring. Hence, the system is not an SDoF system. Assume that the system has 2-DoFs, but the lumped mass at the connection point vanishes. Thus, the dynamic equations of the system satisfy
$$\left{\begin{array}{l} m \frac{\mathrm{d}^2 u_1(t)}{\mathrm{d} t^2}+c\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}\right]=0 \ c\left[\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+k u_2(t)=0 \end{array}\right.$$
Eliminating displacement $u_2$ from Eq. (1.2.1) leads to an ordinary differential equation in terms of velocity $\mathrm{d} u_1(t) / \mathrm{d} t$, namely,
$$\frac{1}{k} \frac{\mathrm{d}^2}{\mathrm{~d} t^2}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{c} \frac{\mathrm{d}}{\mathrm{d} t}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{m}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]=0$$

## 物理代写|振动力学代写振动力学代考|研究的基本思想

$$\left{\begin{array}{l} m \frac{\mathrm{d}^2 u_1(t)}{\mathrm{d} t^2}+c\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}\right]=0 \ c\left[\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+k u_2(t)=0 \end{array}\right.$$

$$\frac{1}{k} \frac{\mathrm{d}^2}{\mathrm{~d} t^2}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{c} \frac{\mathrm{d}}{\mathrm{d} t}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{m}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]=0$$

## MATLAB代写

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