I have this problem (The two rods will be called links. Link 1 has length $a_1$ while link 2 has length $a_2$. The distance of the center of mass of each link to their respective joint is $l_i$):

And I want to find the kinetic energy in order to find the dynamics of the system using the euler-lagrange method. The system has two degrees of freedom ($θ_1,θ_2$). I started by assuming one degree of freedom as a variable and all the others equal to 0.

So the kinetic energy when $θ_2=0, θ_1 \neq 0$ is:

$$T_1 = \frac{1}{2} (I_1+m_1 l_1^2)\dot{θ}_1^2 + \frac{1}{2} \left(I_2+m_2(a_1^2 + l_2^2+2a_1l_2cos(θ_2) \right)\dot{θ}_1^2 $$

Each of this terms has a physical meaning:

1. Rotational energy of $link1$ about joint1: $\frac{1}{2} (I_1+m_1 l_1^2)\dot{θ}_1^2$

2. Rotational energy of $link2$ about joint1: $\frac{1}{2} \left(I_2+m_2(a_1^2 + l_2^2+2a_1l_2cos(θ_2) \right)\dot{θ}_1^2$

The kinetic energy when $θ_1=0, θ_2 \neq 0$ is:

$$ T_2 = \frac{1}{2} (I_2+m_2 l_2^2)\dot{θ}_2^2 $$

which is the rotational energy of $link2$ when rotating about joint 2.

The total kinetic energy is : $T = T_1 + T_2$. However, this is wrong.

If we write the kinetic energy as:

$$ T = \frac{1}{2}m_1|u_1|^2 + \frac{1}{2}I_1ω_1^2 \frac{1}{2}m_2|u_2|^2 \frac{1}{2}Ι_2ω_2^2 $$

and then we write: $$ x_1 = l_1 cos(θ_1), \ y_1 = l_2 sin(θ_1) $$ $$ x_2 = a_1 cos(θ_1) + l_2cos(θ_1+θ_2), \ y_2 = a_2 sin(θ_1)+ l_2sin(θ_1+θ_2) $$

and differentiate this expression, the kinetic energy ends up like that:

$$T = \frac{1}{2} (I_1+m_1 l_1^2)\dot{θ}_1^2 + \frac{1}{2} \left(I_2+m_2(a_1^2 + l_2^2+2a_1l_2cos(θ_2) \right)\dot{θ}_1^2 + \frac{1}{2} (I_2+m_2 l_2^2)\dot{θ}_2^2 + \frac{1}{2} I_2 (2\dot{θ}_1\dot{θ}_2) + \frac{1}{2} m_2(l_2^2+a_1l_2cos(θ_2) )(2\dot{θ}_1\dot{θ}_2) $$ There are two extra terms in that expression:

1. $ \frac{1}{2} I_2 (2\dot{θ}_1\dot{θ}_2) $
2. $\frac{1}{2} m_2(l_2^2+a_1l_2cos(θ_2) )(2\dot{θ}_1\dot{θ}_2) $

The question: What is the physical meaning of these two terms? And could I predict the kinetic energy without writing the explicit expressions of the velocity of each center of mass?

I have this problem (The two rods will be called links. Link 1 has length $a_1$ while link 2 has length $a_2$. The distance of the center of mass of each link to their respective joint is $l_i$):

And I want to find the kinetic energy in order to find the dynamics of the system using the euler-lagrange method. The system has two degrees of freedom ($θ_1,θ_2$). I started by assuming one degree of freedom as a variable and all the others equal to 0.

So the kinetic energy when $θ_2=0, θ_1 \neq 0$ is:

$$T_1 = \frac{1}{2} (I_1+m_1 l_1^2)\dot{θ}_1^2 + \frac{1}{2} \left(I_2+m_2(a_1^2 + l_2^2+2a_1l_2cos(θ_2) \right)\dot{θ}_1^2 $$

Each of this terms has a physical meaning:

1. Rotational energy of $link1$ about joint1: $\frac{1}{2} (I_1+m_1 l_1^2)\dot{θ}_1^2$

2. Rotational energy of $link2$ about joint1: $\frac{1}{2} \left(I_2+m_2(a_1^2 + l_2^2+2a_1l_2cos(θ_2) \right)\dot{θ}_1^2$

The kinetic energy when $θ_1=0, θ_2 \neq 0$ is:

$$ T_2 = \frac{1}{2} (I_2+m_2 l_2^2)\dot{θ}_2^2 $$

which is the rotational energy of $link2$ when rotating about joint 2.

The total kinetic energy is : $T = T_1 + T_2$. However, this is wrong.

If we write the kinetic energy as:

$$ T = \frac{1}{2}m_1|u_1|^2 + \frac{1}{2}I_1ω_1^2 + \frac{1}{2}m_2|u_2|^2 + \frac{1}{2}Ι_2ω_2^2 $$

and then we write: $$ x_1 = l_1 cos(θ_1), \ y_1 = l_2 sin(θ_1) $$ $$ x_2 = a_1 cos(θ_1) + l_2cos(θ_1+θ_2), \ y_2 = a_2 sin(θ_1)+ l_2sin(θ_1+θ_2) $$

and differentiate this expression, the kinetic energy ends up like that:

$$T = \frac{1}{2} (I_1+m_1 l_1^2)\dot{θ}_1^2 + \frac{1}{2} \left(I_2+m_2(a_1^2 + l_2^2+2a_1l_2cos(θ_2) \right)\dot{θ}_1^2 + \frac{1}{2} (I_2+m_2 l_2^2)\dot{θ}_2^2 + \frac{1}{2} I_2 (2\dot{θ}_1\dot{θ}_2) + \frac{1}{2} m_2(l_2^2+a_1l_2cos(θ_2) )(2\dot{θ}_1\dot{θ}_2) $$ There are two extra terms in that expression:

1. $ \frac{1}{2} I_2 (2\dot{θ}_1\dot{θ}_2) $
2. $\frac{1}{2} m_2(l_2^2+a_1l_2cos(θ_2) )(2\dot{θ}_1\dot{θ}_2) $

The question: What is the physical meaning of these two terms? And could I predict the kinetic energy without writing the explicit expressions of the velocity of each center of mass?