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书籍/力学书籍/力学/Flexible Multibody Dynamics (O. A. Bauchau) (Z-Library)/auto/Flexible Multibody Dynamics (O. A. Bauchau) (Z-Library).md
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@ -3123,8 +3123,10 @@ Fig. 3.6. Particle connected to an elastic spring.
Clearly, the situation is similar to that shown in fig. 3.5: the particle is subjected to a central force $\underline{{F}}(r)\,=\,-f(r)\bar{e}_{1}$ . The magnitude of the central force is related to the stretch of the spring, $\varDelta\,=\,r\,-\,r_{0}$ , where $r_{0}$ is the un-stretched length of the spring. For a linearly elastic spring, the force in the spring is proportional to its stretch, $f(r)\,=$ $k(r-r_{0})=k\varDelta$ , where $k$ is the spring stiffness constant. The units of the spring stiffness constant are $\mathrm{N}/\mathrm{m}$ . Clearly, the situation is similar to that shown in fig. 3.5: the particle is subjected to a central force $\underline{{F}}(r)\,=\,-f(r)\bar{e}_{1}$ . The magnitude of the central force is related to the stretch of the spring, $\varDelta\,=\,r\,-\,r_{0}$ , where $r_{0}$ is the un-stretched length of the spring. For a linearly elastic spring, the force in the spring is proportional to its stretch, $f(r)\,=$ $k(r-r_{0})=k\varDelta$ , where $k$ is the spring stiffness constant. The units of the spring stiffness constant are $\mathrm{N}/\mathrm{m}$ .
The potential function of the elastic forces in the spring then follows from eq. (3.27) as 显然，当前情况与图 3.5 所示的情况类似：粒子受到一个中心力 $\underline{{F}}(r)\,=\,-f(r)\bar{e}_{1}$ 。中心力的模值与弹簧的伸长量相关，伸长量定义为 $\varDelta\,=\,r\,-\,r_{0}$ ，其中 $r_{0}$ 是弹簧的未伸长长度。对于线性弹性弹簧，弹簧中的力与它的伸长量成正比，即 $f(r)\,=$ $k(r-r_{0})=k\varDelta$ ，其中 $k$ 是弹簧刚度系数。弹簧刚度系数的单位是 $\mathrm{N}/\mathrm{m}$ 。
The potential function of the elastic forces in the spring then follows from eq. (3.27) as
弹性力势函数则可从公式 (3.27) 推导如下：
$$ $$
V(r)={\frac{1}{2}}k\;\varDelta^{2}. V(r)={\frac{1}{2}}k\;\varDelta^{2}.
$$ $$
@ -3141,10 +3143,22 @@ Next, the set of external forces that maintained the steady deformation $\varDel
Both kinetic and strain energy functions are positive-definite functions, i.e., $K=$ $1/2\:m v^{2}>0$ for any arbitrary speed of the particle $v\neq0$ and $V=1/2\;k\varDelta^{2}>0$ for any stretch of the elastic spring $\varDelta\neq0$ . Consider a strain energy function of the form $\dot{V}\,=\,1/2\,\,k_{0}\varDelta^{2}+1/3\,\,\dot{k}_{1}\varDelta^{3}$ ; this strain energy function vanishes for $\varDelta_{\mathrm{cr}}\,=$ $-3/2\,k_{0}/k_{1}$ . For stretches $\varDelta<\varDelta_{\mathrm{cr}}$ , the strain energy becomes negative, hence this strain energy function is invalid because it is not positive-definite. For $\varDelta<\varDelta_{\mathrm{cr}}$ , the spring will add energy to the system; energy is being created, a physical impossibility for a passive device. Both kinetic and strain energy functions are positive-definite functions, i.e., $K=$ $1/2\:m v^{2}>0$ for any arbitrary speed of the particle $v\neq0$ and $V=1/2\;k\varDelta^{2}>0$ for any stretch of the elastic spring $\varDelta\neq0$ . Consider a strain energy function of the form $\dot{V}\,=\,1/2\,\,k_{0}\varDelta^{2}+1/3\,\,\dot{k}_{1}\varDelta^{3}$ ; this strain energy function vanishes for $\varDelta_{\mathrm{cr}}\,=$ $-3/2\,k_{0}/k_{1}$ . For stretches $\varDelta<\varDelta_{\mathrm{cr}}$ , the strain energy becomes negative, hence this strain energy function is invalid because it is not positive-definite. For $\varDelta<\varDelta_{\mathrm{cr}}$ , the spring will add energy to the system; energy is being created, a physical impossibility for a passive device.
本工作函数通常被称为弹性弹簧的应变能函数。
当前的公式不仅限于线性弹性弹簧：弹簧中的弹性力大小可以是拉伸的非线性函数，例如 $f(r)\,=\,k_{1}\varDelta\,+\,k_{3}\varDelta^{3}$ 。在这种情况下，非线性弹性弹簧的应变能函数为 $V=1/2\;k_{1}\varDelta^{2}+1/4\;k_{3}\varDelta^{4}$ 。
工作与能量原理提供了一种描述粒子动力学的手段，它用能量来代替位移和加速度。考虑图 3.6 所示的系统。在时间 $t_{0}$ 时，粒子处于静止状态，弹簧未拉伸：粒子的速度为零，这意味着 $K_{0}\,=\,0$ ，并且 $V_{0}\,=\,0$ ，因为弹簧未拉伸。施加外部力作用于粒子，使其在时间 $t_{1}$ 时达到新的静止构型，因此 $K_{1}=0$ 。由于系统是保守的，外部力所做的功为 $W_{0\to1}^{\mathrm{ext}}\,=\,E_{1}\,-\,E_{0}\,=\,V_{1}$ 。对于这个简单的例子，工作与能量原理意味着外部施加力所做的功等于弹簧中的应变能。这种功以应变能的形式储存在系统中：没有能量损失，但其性质已从势能转变为应变能。
在这个描述中，粒子从时间 $t_{0}$ 到时间 $t_{1}$ 的轨迹无关紧要；唯一重要的量是弹簧在时间 $t_{1}$ 时的拉伸量 $\varDelta_{1}$ ，它决定了应变能 $V_{1}$ 。这是保守力的一个特征：它们所做的功不取决于从时间 $t_{0}$ 到 $t_{1}$ 遵循的特定路径，而仅仅取决于决定弹簧初始和最终拉伸的系统的初始和最终构型。
接下来，维持弹簧稳态变形 $\varDelta_{1}$ 的外部力系被释放；粒子沿特定轨迹演化，并在时间 $t_{2}$ 时，弹簧的拉伸量为零，$\varDelta_{2}\,=\,0$ 。由于在时间 $t_{1}$ 和 $t_{2}$ 之间没有施加外部力，工作与能量原理意味着 ${\cal W}_{1\to2}^{\mathrm{ext}}\,=\,0\,=\,$ $E_{2}-E_{1}=K_{2}-V_{1}$ ，其中 $K_{2}$ 是粒子在时间 $t_{2}$ 时的动能。没有能量损失：能量从应变能转化为动能，$K_{2}=V_{1}$ 。粒子在时间 $t_{2}$ 时的速度 $v_{2}$ 为 $v_{2}=\sqrt{k/m}\;\varDelta_{1}$ 。再次，粒子所遵循的具体轨迹无关紧要。
动能和应变能函数都是正定函数，即 $K=$ $1/2\:m v^{2}>0$ 对于任何任意速度的粒子 $v\neq0$ 并且 $V=1/2\;k\varDelta^{2}>0$ 对于任何弹簧的拉伸量 $\varDelta\neq0$ 。考虑一种形式的应变能函数 $\dot{V}\,=\,1/2\,\,k_{0}\varDelta^{2}+1/3\,\,\dot{k}_{1}\varDelta^{3}$ ；当 $\varDelta_{\mathrm{cr}}\,=$ $-3/2\,k_{0}/k_{1}$ 时，此应变能函数为零。对于 $\varDelta<\varDelta_{\mathrm{cr}}$ ，应变能变为负，因此此应变能函数无效，因为它不是正定的。对于 $\varDelta<\varDelta_{\mathrm{cr}}$ ，弹簧会向系统添加能量；能量正在被创造，这对于被动设备来说是物理上不可能的。
#### The strain energy function of a torsional spring #### The strain energy function of a torsional spring
Consider the planar problem depicted in fig. 3.7: a particle of mass $m$ is connected to a rigid rod of length $\ell$ . The rod pivots about inertial point O, where a torsional spring of stiffness constant $k$ is located. The torsional spring applies a moment to the rigid rod about point O, which is then transmitted to the particle in the form of a force $\underline{{F}}_{:}$ , acting in the direction normal to the rod; this force is clearly not a central force. The position of the particle will be represented by polar coordinates, $r$ and $\theta$ , see section 2.7.1. The velocity of the particle is $\underline{{v}}\;=\;\dot{r}~\bar{e}_{1}\:+\:r\dot{\theta}~\bar{e}_{2}$ , see eq. (2.91b). Because the rod is rigid, $\dot{r}=0$ , and multiplying the velocity relationship by $\mathrm{d}t$ implies $\underline{{\mathrm{d}}}\underline{{r}}=\ell\mathrm{d}\theta\;\bar{e}_{2}$ . The force vector, $\underline{{F}}$ , has a line of action along $\bar{e}_{2}$ and its magnitude is a function of the sole angle $\theta:\underline{{F}}=-f(\theta)\bar{e}_{2}$ . The differential work done by this force now becomes Consider the planar problem depicted in fig. 3.7: a particle of mass $m$ is connected to a rigid rod of length $\ell$ . The rod pivots about inertial point O, where a torsional spring of stiffness constant $k$ is located. The torsional spring applies a moment to the rigid rod about point O, which is then transmitted to the particle in the form of a force $\underline{{F}}_{:}$ , acting in the direction normal to the rod; this force is clearly not a central force. The position of the particle will be represented by polar coordinates, $r$ and $\theta$ , see section 2.7.1. The velocity of the particle is $\underline{{v}}\;=\;\dot{r}~\bar{e}_{1}\:+\:r\dot{\theta}~\bar{e}_{2}$ , see eq. (2.91b). Because the rod is rigid, $\dot{r}=0$ , and multiplying the velocity relationship by $\mathrm{d}t$ implies $\underline{{\mathrm{d}}}\underline{{r}}=\ell\mathrm{d}\theta\;\bar{e}_{2}$ . The force vector, $\underline{{F}}$ , has a line of action along $\bar{e}_{2}$ and its magnitude is a function of the sole angle $\theta:\underline{{F}}=-f(\theta)\bar{e}_{2}$ . The differential work done by this force now becomes
考虑图 3.7 中所示的平面问题：一个质量为 $m$ 的粒子通过一根长度为 $\ell$ 的刚性杆连接。该杆绕惯性点 O 旋转，惯性点 O 处有一个扭转弹簧，扭转系数为 $k$。扭转弹簧对刚性杆施加一个关于点 O 的扭矩，该扭矩随后以力 $\underline{{F}}_{:}$ 的形式传递给粒子，该力作用于与杆垂直的方向；显然，这并非一种中心力。粒子的位置将用极坐标 $r$ 和 $\theta$ 表示，见第 2.7.1 节。粒子的速度为 $\underline{{v}}\;=\;\dot{r}~\bar{e}_{1}\:+\:r\dot{\theta}~\bar{e}_{2}$ ，见公式 (2.91b)。由于杆是刚性的，$\dot{r}=0$，将速度关系乘以 $\mathrm{d}t$ 得到 $\underline{{\mathrm{d}}}\underline{{r}}=\ell\mathrm{d}\theta\;\bar{e}_{2}$ 。力矢量 $\underline{{F}}$ 的作用线沿 $\bar{e}_{2}$ 方向，其大小是唯一角度 $\theta$ 的函数：$\underline{{F}}=-f(\theta)\bar{e}_{2}$ 。现在，由该力所做的微分功为
![](6fc7ef2a3c96ed9a35e76746df6f58d5374ff193608f00ac2c0c9d4acd500808.jpg) ![](6fc7ef2a3c96ed9a35e76746df6f58d5374ff193608f00ac2c0c9d4acd500808.jpg)
Fig. 3.7. Particle subjected to a force generated by a torsional spring. Fig. 3.7. Particle subjected to a force generated by a torsional spring.
@ -3153,18 +3167,18 @@ $$
$$ $$
Clearly, $M(\theta)=\ell f(\theta)$ is the moment the torsional spring applies to the rigid rod and hence, $\mathrm{d}W=-M(\theta)\mathrm{d}\theta$ . For a linearly elastic torsional spring, $M(\theta)=k(\theta-\theta_{0})$ , where $\theta_{0}$ is the angular position of the rigid rod for which the torsional spring is unstretched. The units for the stiffness constant $k$ are N·m/rad. The potential function for the torsional spring now becomes Clearly, $M(\theta)=\ell f(\theta)$ is the moment the torsional spring applies to the rigid rod and hence, $\mathrm{d}W=-M(\theta)\mathrm{d}\theta$ . For a linearly elastic torsional spring, $M(\theta)=k(\theta-\theta_{0})$ , where $\theta_{0}$ is the angular position of the rigid rod for which the torsional spring is unstretched. The units for the stiffness constant $k$ are N·m/rad. The potential function for the torsional spring now becomes
显然，$M(\theta)=\ell f(\theta)$ 是扭转弹簧施加在刚性杆上的力矩，因此，$\mathrm{d}W=-M(\theta)\mathrm{d}\theta$。对于线性弹性扭转弹簧，$M(\theta)=k(\theta-\theta_{0})$，其中 $\theta_{0}$ 是刚性杆的角位置，此时扭转弹簧处于无拉伸状态。刚度常数 $k$ 的单位是 N·m/rad。扭转弹簧的势函数现在变为
$$ $$
V(\theta)=\frac{1}{2}\;k(\theta-\theta_{0})^{2}. V(\theta)=\frac{1}{2}\;k(\theta-\theta_{0})^{2}.
$$ $$
This potential function is called the strain energy function of the torsional spring. It is also possible to define nonlinearly elastic torsional springs, for which the elastic moment is a nonlinear function of angle $\theta$ ; for instance, if $M(\theta)~~=~~k_{1}(\theta~-~\theta_{0})~+~k_{3}(\theta~-~\theta_{0})^{3}$ , the strain energy function is then This potential function is called the strain energy function of the torsional spring. It is also possible to define nonlinearly elastic torsional springs, for which the elastic moment is a nonlinear function of angle $\theta$ ; for instance, if $M(\theta)~~=~~k_{1}(\theta~-~\theta_{0})~+~k_{3}(\theta~-~\theta_{0})^{3}$ , the strain energy function is then
这个势函数被称为扭转弹簧的应变能函数。 也可以定义非线性弹性扭转弹簧，其弹性弯矩是角度 $\theta$ 的非线性函数；例如，如果 $M(\theta)~~=~~k_{1}(\theta~-~\theta_{0})~+~k_{3}(\theta~-~\theta_{0})^{3}$ ，那么应变能函数则为
$$ $$
V(\theta)=k_{1}(\theta-\theta_{0})^{2}/2+k_{3}(\theta-\theta_{0})^{4}/4. V(\theta)=k_{1}(\theta-\theta_{0})^{2}/2+k_{3}(\theta-\theta_{0})^{4}/4.
$$ $$
# 3.2.3 Non-conservative forces ### 3.2.3 Non-conservative forces
![](9cda79e95b811606ba414f470fe19af768d2fc51dfe0b94c3e06faa5dcc032e1.jpg) ![](9cda79e95b811606ba414f470fe19af768d2fc51dfe0b94c3e06faa5dcc032e1.jpg)
Fig. 3.8. Particle connected to a dashpot. Fig. 3.8. Particle connected to a dashpot.
@ -3177,6 +3191,13 @@ If the position of the particle is expressed in terms of spherical coordinates,
The work done by the viscous forces in the dashpot is The work done by the viscous forces in the dashpot is
现在考虑一个质量为 $m$ 的粒子，它通过一个直线阻尼器连接；阻尼器的另一端连接到一个惯性点 $\mathbf{o}$ ，如图 3.8 所示。阻尼器可以轴向滑动，并且质量可以忽略不计；实际上，这意味着它的质量与粒子的质量相比可以忽略不计。
对于线性阻尼器，它产生的粘性力的大小与它的长度随时间的变化率成正比，即 $f({\dot{r}})=c{\dot{r}}$ 。系数 $c$ 被称为阻尼系数，其单位是 $\mathrm{N}{\cdot}\mathrm{s}/\mathrm{m}$ 。 尽管阻尼器产生的力的作用线通过惯性点，但它不是一个中心力，因为它的大小不取决于粒子和惯性点之间的距离。
如果粒子的位置用球坐标表示，那么微分位移就是 $\mathrm{d}\underline{{r}}=\mathrm{d}r\;\bar{e}_{1}+r\mathrm{d}\phi\;\bar{e}_{2}+r\mathrm{d}\theta\sin\phi\;\bar{e}_{3}$ 。阻尼器力所做的微分功现在变为 $\mathrm{d}W=\underline{{F}}^{T}\mathrm{d}\underline{{r}}=-f(\dot{r})\bar{e}_{1}^{T}(\mathrm{d}r\ \bar{e}_{1}+$ $r\mathrm{d}\phi\,\bar{e}_{2}+r\mathrm{d}\theta\sin\phi\,\bar{e}_{3})=-c\dot{r}\,\mathrm{d}r$ 。 由于粘性力依赖于 $\dot{r}$，微分功不能写成一个全微分的形式；不存在一个势函数 $V(r)$，使得 $\mathrm{d}V/\mathrm{d}r=-c\dot{r}\mathrm{d}r$ 。阻尼器中的力是一个非保守力。
阻尼器中粘性力所做的功是
$$ $$
W_{t_{i}\to t_{f}}=-\int_{t_{i}}^{t_{f}}c{\dot{r}}\,\mathrm{d}r=-\int_{t_{i}}^{t_{f}}c{\dot{r}}{\frac{\mathrm{d}r}{\mathrm{d}t}}\,\mathrm{d}t=-\int_{t_{i}}^{t_{f}}c{\dot{r}}^{2}\,\mathrm{d}t<0. W_{t_{i}\to t_{f}}=-\int_{t_{i}}^{t_{f}}c{\dot{r}}\,\mathrm{d}r=-\int_{t_{i}}^{t_{f}}c{\dot{r}}{\frac{\mathrm{d}r}{\mathrm{d}t}}\,\mathrm{d}t=-\int_{t_{i}}^{t_{f}}c{\dot{r}}^{2}\,\mathrm{d}t<0.
$$ $$
@ -3186,6 +3207,11 @@ The presence of the $\dot{r}^{2}$ term implies that the work done by the viscous
Of course, dashpots are not always linear; the magnitude of the viscous force could be a nonlinear function of velocity, such as $f(\dot{r})=c_{1}\dot{r}+c_{3}\dot{r}^{3}$ , for instance. Function $f(\dot{\boldsymbol{r}})\dot{\boldsymbol{r}}$ , however, must be a positive-definite function of $\dot{r}$ to guarantee the dissipative nature of the resulting viscous force. Of course, dashpots are not always linear; the magnitude of the viscous force could be a nonlinear function of velocity, such as $f(\dot{r})=c_{1}\dot{r}+c_{3}\dot{r}^{3}$ , for instance. Function $f(\dot{\boldsymbol{r}})\dot{\boldsymbol{r}}$ , however, must be a positive-definite function of $\dot{r}$ to guarantee the dissipative nature of the resulting viscous force.
Finally, it is also possible to encounter torsional dashpots; in fig. 3.7, the torsional spring would be replaced by a dashpot that applies to the rigid bar a moment whose magnitude is a function of the time rate of change of angle $\theta$ . The differential work done by the viscous forces in the torsional is then $\mathrm{d}W=-\ell f(\dot{\theta})\mathrm{d}\theta=-M(\dot{\theta})\mathrm{d}\theta$ ; for a linear torsional dashpot, $M({\dot{\theta}})=c{\dot{\theta}}$ , where the dashpot constant now has units of $\mathbf{N}{\cdot}\mathbf{m}{\cdot}\mathbf{s}$ . Finally, it is also possible to encounter torsional dashpots; in fig. 3.7, the torsional spring would be replaced by a dashpot that applies to the rigid bar a moment whose magnitude is a function of the time rate of change of angle $\theta$ . The differential work done by the viscous forces in the torsional is then $\mathrm{d}W=-\ell f(\dot{\theta})\mathrm{d}\theta=-M(\dot{\theta})\mathrm{d}\theta$ ; for a linear torsional dashpot, $M({\dot{\theta}})=c{\dot{\theta}}$ , where the dashpot constant now has units of $\mathbf{N}{\cdot}\mathbf{m}{\cdot}\mathbf{s}$ .
$\dot{r}^{2}$ 项的存在意味着粘性力所做的功始终为负，即它们是耗散力。对于图 3.8 中所示的系统，工作与能量原理表明 $W_{t_{i}\rightarrow t_{f}}\,=\,E_{f}\,-\,E_{i}$ ，或者 $E_{f}\,=\,E_{i}\,+$ $W_{t_{i}\to t_{f}}$ 。由于功为负量，系统的总机械能随时间单调递减；此外，总机械能的变化正好等于阻尼器中粘性力所做的功。这一结果解释了用来描述阻尼器中粘性力的“耗散力”或“非保守力”一词。
当然，阻尼器并非总是线性的；粘性力的幅度可能是速度的非线性函数，例如 $f(\dot{r})=c_{1}\dot{r}+c_{3}\dot{r}^{3}$ 。然而，函数 $f(\dot{\boldsymbol{r}})\dot{\boldsymbol{r}}$ 必须是 $\dot{r}$ 的正定函数，以保证由此产生的粘性力的耗散特性。
最后，也可能遇到扭转阻尼器；在图 3.7 中，扭转弹簧将被一个阻尼器取代，该阻尼器对刚性杆施加一个力矩，其大小是角度 $\theta$ 随时间变化率的函数。扭转阻尼器中粘性力所做的微分功为 $\mathrm{d}W=-\ell f(\dot{\theta})\mathrm{d}\theta=-M(\dot{\theta})\mathrm{d}\theta$；对于线性扭转阻尼器，$M({\dot{\theta}})=c{\dot{\theta}}$ ，其中阻尼器常数的单位现在为 $\mathbf{N}{\cdot}\mathbf{m}{\cdot}\mathbf{s}$ 。
# The energy closure equation # The energy closure equation