laws-of-motion

Question

hi there ,can you pls explain me how to write constraint equation

Aman Bansal · Accepted Answer

Dear Sivaraman,

n the study of the dynamics of mechanical systems, the configuration of a given system $S$ is, in general, completely described by $n$ generalized coordinates so that its generalized coordinate $n$ -vector is given by

$q:=[q_1,q_2,\ldots,q_n]^T.$

Using Newtonian or Lagrangian dynamics, the unconstrained equations of motion of the system $S$ under study can be derived as

$M(q,t)\ddot{q}(t)=Q(q,\dot{q},t),$

where it is assumed that the initial conditions $q(0)$ and $\dot{q}(0)$ are known. We call the system $S$ unconstrained because $\dot{q}(0)$ may be arbitrarily assigned. Here, the dots represent derivatives with respect to time. The $n$ by $n$ matrix $M$ is symmetric, and it can be positive definite $(M > 0)$ or semi-positive definite $(M \geq 0)$ . Typically, it is assumed that $M$ is positive definite; however, it is not uncommon to derive the unconstrained equations of motion of the system $S$ such that $M$ is only semi-positive definite; i.e., the mass matrix may be singular.^[2]^[3] The $n$ -vector $Q$ denotes the total generalized force impressed on the system; it can be expressible as the summation of all the conservative forces with the non-conservative forces.

Constraints

We now assume that the unconstrained system $S$ is subjected to a set of $m$ consistent equality constraints given by

$A(q,\dot{q},t)\ddot{q} = b(q,\dot{q},t),$

where $A$ is a known m by n matrix of rank r and $b$ is a known m-vector. We note that this set of constraint equations encompass a very general variety of holonomic and non-holonomic equality constraints. For example, holonomic constraints of the form

$\varphi(q,t) = 0$

can be differentiated twice with respect to time while non-holonomic constraints of the form

$\psi(q,\dot{q},t) = 0$

can be differentiated once with respect to time to obtain the $m$ by $n$ matrix $A$ and the $m$ -vector $b$ . In short, constraints may be specified that are (1) nonlinear functions of displacement and velocity, (2) explicitly dependent on time, and (3) functionally dependent.

As a consequence of subjecting these constraints to the unconstrained system $S$ , an additional force is conceptualized to arise, namely, the force of constraint. Therefore, the constrained system $S_c$ becomes

$M\ddot{q}=Q+Q^{c}(q,\dot{q},t),$

where $Q^{c}$ —the constraint force—is the additional force needed to satisfy the imposed constraints. The central problem of constrained motion is now stated as follows:

1. given the unconstrained equations of motion of the system $S$ ,

2. given the generalized displacement $q(t)$ and the generalized velocity $\dot{q}(t)$ of the constrained system $S_c$ at time $t$ , and

3. given the constraints in the form $A\ddot{q}=b$ as stated above,

find the equations of motion for the constrained system—the acceleration—at time t, which is in accordance with the agreed upon principles of analytical dynamics.

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General Physics> laws-of-motion...

hi there ,can you pls explain me how to write constraint equation

sivaraman sankar , 13 Years ago

Aman Bansal

Constraints

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General Physics> laws-of-motion...

hi there ,can you pls explain me how to write constraint equation

sivaraman sankar , 13 Years ago

Aman Bansal

Constraints

LIVE ONLINE CLASSES

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