Akademia Górniczo-Hutnicza

w Krakowie

Modelowanie w Projektowaniu Maszyn

Projekt

Stół wibracyjny

Hyrchel Marcin

Mazan Maciej

Nowak Patryk

Padło Maciej

Szlachta Grzegorz

Gr. K2

1. Dane

f = 0, 23 [m]	N = 1, 5 [kW]
e = 0, 025 [m]	nN = 1415 [obr]
b = 1, 5 [m]	n0 = 1500 [obr]
a = 0, 45[m] $$l = \frac{b}{2} = 0,75\left\lbrack m \right\rbrack$$ $$h = \frac{a}{2} = 0,225\left\lbrack m \right\rbrack$$	przeciazalnosc p = 3
M = 112 [kg]	n = 2 lb. okresow drgan
ml = 15, 2 [kg]	T = 0, 12 [s] okres drgan
mw = 7 [kg]	A1 = 7, 6 [mm]
J = 22, 89 [kg * m^2]	A3 = 1, 2 [mm]
Jl = 0, 3 [kg * m^2]
JwA = 0, 0285 [kg * m^2]
Jw = JwA + e^2 * mw  = 0, 033[kg * m^2]

1. Współrzędne uogólnione

Przyjęte współrzędne uogólnione: {x,y,φ,β,α}
Założenia:
sinβ = β sinφ = φ cosβ = 1 cosφ = 1

1. Współrzędne wierzchołków

xA = x − lsinφ xB = x − lsinφ xC = x + lsinφ xD = x + lsinφ	yA = y + hsinφ yB = y − hsinφ yC = y − hsinφ yD = y + hsinφ

xA = x − lφ xB = x − lφ xC = x + lφ xD = x + lφ	yA = y + hφ yB = y − hφ yC = y − hφ yD = y + hφ

1. Równania więzów

$$\left\{ \begin{matrix} x_{w} = ecos \propto + fcos\beta + x + lcos\varphi \\ y_{w} = esin\alpha + fsin\beta + y + lsin\varphi \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} x_{w} = ecos\alpha + f + x + l \\ y_{w} = esin\alpha + f\beta + y + l\varphi \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} {\dot{x}}_{w} = - e\dot{\alpha}sin\alpha + \dot{x} \\ {\dot{y}}_{w} = e\dot{\alpha}cos\alpha + f\dot{\beta} + \dot{y} + l\dot{\varphi} \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} x_{l} = fcos\beta + x + lcos\varphi \\ y_{l} = fsin\beta + y + lsin\varphi \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} x_{l} = f + x + l \\ y_{l} = f\beta + y + l\varphi \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} {\dot{x}}_{l} = \dot{x} \\ {\dot{y}}_{l} = f\dot{\beta} + \dot{y} + l\dot{\varphi} \\ \end{matrix} \right.\ $$

1. Obliczanie

• Moment elektryczny silnika

$$M_{n} = 9550*\frac{N}{n_{N}} = 10,12\ \text{Nm}$$

$$\omega_{N} = \frac{\pi n_{N}}{30} = 148,2\ \frac{1}{s}$$

M_ut = p * M_n = 3 * 10 = 30, 36 Nm

$$\omega_{s} = \frac{\pi n_{s}}{30} = 157,1\ \frac{\text{rad}}{s}$$

$$s_{N} = \frac{\omega_{s} - \omega_{N}}{\omega_{s}} = \frac{157,1 - 148,2}{157,1} = 0,057$$

$$s_{\text{ut}} = s_{N}\left( p + \sqrt{p^{2} - 1} \right) = 0,057*\left( 3 + \sqrt{3^{2} - 1} \right) = 0,33$$

$$\omega_{\text{ut}} = \omega_{s}\left( 1 - s_{\text{ut}} \right) = 105,3\frac{\text{rad}}{s}$$

$$M_{\text{el}} = \frac{2M_{\text{ut}}(\omega_{s} - \omega_{\text{ut}})(\omega_{s} - \omega_{\alpha})}{\left( \omega_{s} - \omega_{\text{ut}} \right)^{2} + {(\omega_{s} - \omega_{\alpha})}^{2}} = \frac{2*30,36*(157,1 - 105,3)(157,1 - \omega_{\alpha})}{{(157,1 - 105,3)}^{2} + {(157,1 - \omega_{\alpha})}^{2}} = \frac{3145,3*(157,1 - \omega_{\alpha})}{2683,2*{(157,1 - \omega_{\alpha})}^{2}}$$

$$\omega_{\alpha} = \dot{\alpha}$$

• Współczynnik tłumienia

$$b = \frac{2\text{mδ}}{\text{nT}}\ \left\lbrack \frac{\text{kg}}{s} \right\rbrack$$

Dla 4 tłumików

$$m = \frac{M + m_{l} + m_{w}}{4} = \frac{112 + 15,2 + 7}{4} = 33,55\ \lbrack\text{kg}\rbrack$$

$$\delta = \ln\left\lbrack \frac{x(t_{0})}{x(t_{0} + \text{nT})} \right\rbrack = \ln\frac{7}{1,2} = 1,7636$$

$$b = \frac{2\text{mδ}}{\text{nT}} = \frac{2*33,55*1,7636}{2*0,12} = 493,1\ \left\lbrack \frac{\text{kg}}{s} \right\rbrack$$

• Współczynnik sztywności sprężyny

$$\omega_{tl} = \frac{2\pi}{T} = \frac{\sqrt{4\text{km} - b^{2}}}{2m}$$

$$k = \frac{4m\pi^{2}}{T^{2}} + \frac{b^{2}}{4m} = \frac{4*33,55*{3,14}^{2}}{{0,12}^{2}} + \frac{{493,1}^{2}}{4*33,55} = 93697\ \frac{N}{m}$$

1. Energia kinetyczna

$$E_{k} = \frac{1}{2}*M*{\dot{x}}^{2} + \frac{1}{2}*M*{\dot{y}}^{2} + \frac{1}{2}*J*{\dot{\varphi}}^{2} + \frac{1}{2}m_{l}\ {\dot{x}}^{2} + \frac{1}{2}m_{l}\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right)^{2} + \frac{1}{2}J_{l}{\dot{\beta}}^{2} + \frac{1}{2}m_{w}\left( \dot{x} - e\dot{\alpha}\text{sinα} \right)^{2} + \frac{1}{2}m_{w}\left( \dot{y} + l\dot{\varphi} + f\dot{\beta} + e\dot{\alpha}\text{cosα} \right)^{2} + \frac{1}{2}J_{w}{\dot{\alpha}}^{2}$$

1. Energia potencjalna

$$E_{p} = \frac{1}{2}*k\left( 4x^{2} + 4l^{2}\varphi^{2} \right) + \frac{1}{2}*k\left( 4y^{2} + {4h}^{2}\varphi^{2} \right) = 2kx^{2} + 2kl^{2}\varphi^{2} + 2ky^{2} + {2kh}^{2}\alpha^{2}$$

1. Potencjał Lagrange’a

L = E_k − E_p

$$L = \frac{1}{2}M{\dot{x}}^{2} + \frac{1}{2}M{\dot{y}}^{2} + \frac{1}{2}J{\dot{\varphi}}^{2} + \frac{1}{2}m_{l}\ {\dot{x}}^{2} + \frac{1}{2}m_{l}\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right)^{2} + \frac{1}{2}J_{l}{\dot{\beta}}^{2} + \frac{1}{2}m_{w}\left( \dot{x} - e\dot{\alpha}\text{sinα} \right)^{2} + \frac{1}{2}m_{w}\left( \dot{y} + l\dot{\varphi} + f\dot{\beta} + e\dot{\alpha}\text{cosα} \right)^{2} + \frac{1}{2}J_{w}{\dot{\alpha}}^{2} - (2kx^{2} + 2kl^{2}\varphi^{2} + 2ky^{2} + {2kh}^{2}\alpha^{2})$$

1. Moc strat (dyssypacja energii)

$$N = b{\dot{x}}^{2} + b{\dot{y}}^{2} = b\left( 4{\dot{x}}^{2} + {4l}^{2}{\dot{\varphi}}^{2} \right) + b\left( 4{\dot{y}}^{2} + {4h}^{2}{\dot{\varphi}}^{2} \right) = 4b{\dot{x}}^{2} + 4b{\dot{y}}^{2} + {\dot{\varphi}}^{2}\left( {4bl}^{2} + {4bh}^{2} \right)$$

1. Równania różniczkowe dla współrzędnych uogólnionych

dla x

$$Q_{x} = \frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{x}} - \frac{\partial L}{\partial x} + \frac{1}{2}\frac{\partial N}{\partial\dot{x}} = 0$$

$$\frac{\partial L}{\partial\dot{x}} = M\dot{x} + m_{l}\ \dot{x} + m_{w}\left( \dot{x} - e\dot{\alpha}\text{sinα} \right)\backslash n$$

$$\frac{\partial L}{\partial x} = - 4\text{kx}$$

$$\frac{\partial N}{\partial\dot{x}} = 8b\dot{x}$$

$$Q_{x} = \ddot{x}\left( M + m_{l} + m_{w} \right) + m_{w}( - e\ddot{\alpha}sin\alpha - e{\dot{\alpha}}^{2}cos\alpha) + 4\text{kx} + 4b\dot{x} = 0$$

dla y

$$Q_{y} = \frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{y}} - \frac{\partial L}{\partial y} + \frac{1}{2}\frac{\partial N}{\partial\dot{y}}$$

$$\frac{\partial L}{\partial\dot{y}} = M\dot{y} + m_{l}\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right) + m_{w}(\dot{y} + f\dot{\beta} + l\dot{\varphi} + e\dot{\alpha}cos\alpha)$$

$$\frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{y}} = M\ddot{y} + m_{l}\left( \ddot{y} + f\ddot{\beta} + l\ddot{\varphi} \right) + m_{w}\left( \ddot{y} + f\ddot{\beta} + l\ddot{\varphi} + e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}\text{sinα} \right) = \ddot{y}\left( M + m_{l} + m_{w} \right) + \ddot{\varphi}l\left( m_{l} + m_{w} \right) + \ddot{\beta}f\left( m_{l} + m_{w} \right) + m_{w}(e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}sin\alpha)$$

$$\frac{\partial L}{\partial y} = - 4\text{ky}$$

$$\frac{\partial N}{\partial\dot{y}} = 8b\dot{y}$$

$$Q_{y} = \ddot{y}\left( M + m_{l} + m_{w} \right) + \left( \ddot{\varphi}l + \ddot{\beta}f \right)\left( m_{l} + m_{w} \right) + m_{w}\left( e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}\text{sinα} \right) + 4ky + 4b\dot{y}$$

dla φ

$$Q_{\varphi} = \frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{\varphi}} - \frac{\partial L}{\partial\varphi} + \frac{1}{2}\frac{\partial N}{\partial\dot{\varphi}}$$

$$\frac{\partial L}{\partial\dot{\varphi}} = J\dot{\varphi} + m_{l}l\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right) + m_{w}l(\dot{y} + f\dot{\beta} + l\dot{\varphi} + e\dot{\alpha}cos\alpha)$$

$$\frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{\varphi}} = J\ddot{\varphi} + m_{l}l\left( \ddot{y} + f\ddot{\beta} + l\ddot{\varphi} \right) + m_{w}l(\ddot{y} + f\ddot{\beta} + l\ddot{\varphi} + e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}sin\alpha)$$

$$\frac{\partial L}{\partial\varphi} = - 4kh^{2}\varphi - 4kl^{2}\varphi$$

$$\frac{\partial N}{\partial\dot{\varphi}} = 8bh^{2}\dot{\varphi} + 8bl^{2}\dot{\varphi}$$

$$Q_{\varphi} = \ddot{\varphi}\ \left( J + m_{l}l^{2} + m_{w}l^{2} \right) + \left( \ddot{y} + f\ddot{\beta} \right)\left( m_{l}l + m_{w}l \right) + m_{w}l\left( e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}\text{sinα} \right) + 4kh^{2}\varphi - 4kl^{2}\varphi + 4bh^{2}\dot{\varphi} + 4bl^{2}\dot{\varphi}$$

dla β

$$Q_{\beta} = \frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{\beta}} - \frac{\partial L}{\partial\beta} + \frac{1}{2}\frac{\partial N}{\partial\dot{\beta}}$$

$$\frac{\partial L}{\partial\dot{\beta}} = m_{l}f\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right) + J_{l}\dot{\beta} + m_{w}f(\dot{y} + f\dot{\beta} + l\dot{\varphi} + e\dot{\alpha}cos\alpha)$$

$$\frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{\beta}} = \ddot{\beta}\ \left( J_{l} + m_{l}f^{2} + m_{w}f^{2} \right) + \left( \ddot{y} + l\ddot{\varphi} \right)\left( m_{l}f + m_{w}f \right) + m_{w}f\left( e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}\text{sinα} \right)$$

$$\frac{\partial L}{\partial\beta} = 0$$

$$\frac{\partial N}{\partial\dot{\beta}} = 0$$

$$Q_{\beta} = \ddot{\beta}\ \left( J_{l} + m_{l}f^{2} + m_{w}f^{2} \right) + \left( \ddot{y} + l\ddot{\varphi} \right)\left( m_{l}f + m_{w}f \right) + m_{w}f\left( e\ddot{\alpha}cos\alpha - e{\dot{\alpha}}^{2}\text{sinα} \right) = 0$$

dla α

$$Q_{\alpha} = \frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{\alpha}} - \frac{\partial L}{\partial\alpha} + \frac{1}{2}\frac{\partial N}{\partial\dot{\alpha}}$$

$$\frac{\partial L}{\partial\dot{\alpha}} = - m_{w}\text{esin}\alpha\left( \dot{x} - e\dot{\alpha}\sin\alpha \right){+ m}_{w}\text{ecos}\alpha\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} + e\dot{\alpha}\text{cosα} \right) + J_{w}\dot{\alpha} = m_{w}e^{2}\dot{\dot{\alpha}(\sin^{2}\alpha + \cos^{2}\alpha)} + J_{w}\dot{\alpha} - m_{w}\text{esinα}\dot{x} + m_{w}\text{ecos}\alpha\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right) = \dot{\alpha}(m_{w}e^{2} + J_{w}) - m_{w}e\dot{x}\text{sinα} + m_{w}\text{ecos}\alpha\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} \right)$$

$$\frac{d}{\text{dt}}\frac{\partial L}{\partial\dot{\alpha}} = \ddot{\alpha}(m_{w}e^{2} + J_{w}) - m_{w}e(\ddot{x}sin\alpha + \dot{x}\dot{\alpha}\cos\alpha - \ddot{y}\cos\alpha + \dot{y}\dot{\alpha}cos\alpha - f\ddot{\beta}\cos\alpha + f\dot{\beta}\dot{\alpha}\sin\alpha - l\ddot{\varphi}cos\alpha + l\dot{\varphi}\dot{\alpha}sin\alpha)$$

$$\frac{\partial L}{\partial\alpha} = m_{w}e\dot{\alpha}\cos\alpha\left( \dot{x} - e\dot{\alpha}\sin\alpha \right){- m}_{w}e\dot{\alpha}\sin\alpha\left( \dot{y} + f\dot{\beta} + l\dot{\varphi} + e\dot{\alpha}\text{cosα} \right) = m_{w}e\dot{\alpha}(\dot{x}\cos\alpha - \dot{y}\sin\alpha - f\dot{\beta}\sin\alpha - l\dot{\varphi}\sin\alpha)$$

$$\frac{\partial N}{\partial\dot{\varphi}} = 0$$

$$Q_{\alpha} = \ddot{\alpha}(m_{w}e^{2} + J_{w}) - m_{w}e(\ddot{x}sin\alpha + \dot{x}\dot{\alpha}\cos\alpha - \ddot{y}\cos\alpha + \dot{y}\dot{\alpha}cos\alpha - f\ddot{\beta}\cos\alpha + f\dot{\beta}\dot{\alpha}\sin\alpha - l\ddot{\varphi}cos\alpha + l\dot{\varphi}\dot{\alpha}sin\alpha) - m_{w}e\dot{\alpha}(\dot{x}\cos\alpha - \dot{y}\sin\alpha - f\dot{\beta}\sin\alpha - l\dot{\varphi}\sin\alpha) = M_{\text{el}}$$

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