OpenGL - Translate an Object on a arbitrary axis - c++

Question

I have a sphere and also the axis through its origin. I want to translate the sphere on its axis up and down. In it's starting position that's no problem, as the axis start parallel to the global y axis., but as soon as I rotate the sphere and therefore also the sphere axis around the z-axis it gets complicated.
My first thought was to normalize the axis and simply use that as the translation vector in the translation matrix. Then multiply the translation matrix with the normalized axis and the sphere should be pushed one unit on the axis.
Here is the code I already got:
class Object
{
public:
inline Object()
: vao(0),
positionBuffer(0),
colorBuffer(0),
indexBuffer(0),
elements(0),
vertices(0)
{}
inline ~Object() { // GL context must exist on destruction
glDeleteVertexArrays(1, &vao);
glDeleteBuffers(1, &indexBuffer);
glDeleteBuffers(1, &colorBuffer);
glDeleteBuffers(1, &positionBuffer);
}
GLuint vao; // vertex-array-object ID
GLuint positionBuffer; // ID of vertex-buffer: position
GLuint colorBuffer; // ID of vertex-buffer: color
GLuint indexBuffer; // ID of index-buffer
GLuint elements; // Number of Elements
vector<glm::vec3> vertices;
glm::vec3 mp;
glm::mat4x4 model; // model matrix
};
glm::vec3 axis = glm::normalize(glm::vec3{
sphereax.vertices[0].x - sphereax.vertices[1].x,
sphereax.vertices[0].y - sphereax.vertices[1].y,
sphereax.vertices[0].z - sphereax.vertices[1].z}
);
translateObject(earth, axis);
void translateObject(Object &obj, glm::vec3 &translation)
{
glm::mat4x4 trans_mat = glm::translate(glm::mat4(1.0f), translation);
for (int i = 0; i < obj.vertices.size(); i++)
{
obj.vertices[i] = glm::vec3(glm::vec4(obj.vertices[i], 1.0f) * trans_mat);
}
obj.mp = glm::vec3(glm::vec4(obj.mp, 1.0f) * trans_mat);
}
The translation matrix in translateObject(); seems to be right, yet the multiplication of one of the points and the transformation matrix shows no effect.

Answer 1

I suggest to do the following:
Rotate the sphere around the its rotation axis
Translate the sphere by its distance form the center along the x axis
Rotate the sphere around the center of the world
glm::vec3 rotation_axis = ...; // sphere axis
float axis_rot_angle = ...; // current axis rotation angle in radians
float distance_to_center = ...; // distance from the center of the world
float world_rot_angle = ...; // current world rotation angle in radians
glm::mat4 sphere_rot = glm::rotation(glm::mat4(1.0f), axis_rot_angle, rotation_axis);
glm::mat4 sphere_trans = glm::translate(glm::mat4(1.0f), glm::vec3(distance_to_center, 0.0f, 0.0f));
glm::mat4 world_rot = glm::rotation(glm::mat4(1.0f), world_rot_angle, glm::vec3(0.0f, 0.0f, 1.0f));
glm::mat4 model_mat = world_rot * sphere_trans * sphere_rot;